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**PHILADELPHIA, Pennsylvania, May 8th, 2024/CyberNewsWire/--**Security Risk Advisors (SRA) announces the launch of their OT/XIoT Detection Selection Workshop, a complimentary offering designed to assist organizations in selecting the most suitable operational technology (OT) and Extended Internet of Things (XIoT) security tools for their unique environments. Led by seasoned OT/XIoT security consultants, the workshop provides participants with an invaluable opportunity to gain insights into both best-in-class and novel solutions and to identify those closest aligned to their specific needs .In today’s increasingly interconnected digital landscape, the importance of choosing the right OT/XIoT security tools cannot be overstated. These tools serve as the first line of risk reduction and defense against cyber threats targeting critical industrial processes and infrastructure. Making informed decisions, whether adding a new solution or replacing an incumbent, significantly impacts an organization’s ability to mitigate threats and protect its assets. During the half-day consultation, participants will delve deep into their OT/XIoT security environments, examining current tools and analyzing their infrastructure. The free workshop will result in personalized recommendations of the best-fit solutions from industry vendors. “We recognize the importance of selecting the right security tools for cyber-physical environments,” says Jason Rivera, Director of OT/XIoT Security at SRA. “Our workshop empowers organizations to make informed decisions, giving confidence that their selection is fit for purpose.” Submit your application here. About Security Risk Advisors Security Risk Advisors offers Purple Teams, Cloud Security, Penetration Testing, OT Security, and 24x7x365 Cybersecurity Operations. Based in Philadelphia, SRA operates across the USA, Ireland and Australia. For more information, visit SRA’s website at https://sra.io. Contact Marketing Manager Douglas Webster news@sra.io 215-867-9051 This story was distributed as a release by Cyberwire under HackerNoon’s Business Blogging Program. Learn more about the program here. **PHILADELPHIA, Pennsylvania, May 8th, 2024/CyberNewsWire/--**Security Risk Advisors (SRA) announces the launch of their OT/XIoT Detection Selection Workshop, a complimentary offering designed to assist organizations in selecting the most suitable operational technology (OT) and Extended Internet of Things (XIoT) security tools for their unique environments. Led by seasoned OT/XIoT security consultants, the workshop provides participants with an invaluable opportunity to gain insights into both best-in-class and novel solutions and to identify those closest aligned to their specific needs .In today’s increasingly interconnected digital landscape, the importance of choosing the right OT/XIoT security tools cannot be overstated. These tools serve as the first line of risk reduction and defense against cyber threats targeting critical industrial processes and infrastructure. Making informed decisions, whether adding a new solution or replacing an incumbent, significantly impacts an organization’s ability to mitigate threats and protect its assets. During the half-day consultation, participants will delve deep into their OT/XIoT security environments, examining current tools and analyzing their infrastructure. The free workshop will result in personalized recommendations of the best-fit solutions from industry vendors. “We recognize the importance of selecting the right security tools for cyber-physical environments,” says Jason Rivera, Director of OT/XIoT Security at SRA. “Our workshop empowers organizations to make informed decisions, giving confidence that their selection is fit for purpose.” “We recognize the importance of selecting the right security tools for cyber-physical environments,” says Jason Rivera, Director of OT/XIoT Security at SRA. “Our workshop empowers organizations to make informed decisions, giving confidence that their selection is fit for purpose.” Submit your application here. Submit your application here. Submit your application here. About Security Risk Advisors Security Risk Advisors offers Purple Teams, Cloud Security, Penetration Testing, OT Security, and 24x7x365 Cybersecurity Operations. Based in Philadelphia, SRA operates across the USA, Ireland and Australia. For more information, visit SRA’s website at https://sra.io . https://sra.io https://sra.io Contact Marketing Manager Douglas Webster news@sra.io 215-867-9051 This story was distributed as a release by Cyberwire under HackerNoon’s Business Blogging Program. Learn more about the program here. This story was distributed as a release by Cyberwire under HackerNoon’s Business Blogging Program. Learn more about the program here . here here

Free Workshop From Security Risk Advisors Empowers Organizations To Select Optimal OT Security Tools

নিরাপত্তা ঝুঁকি উপদেষ্টাদের থেকে বিনামূল্যে কর্মশালা সর্বোত্তম OT নিরাপত্তা সরঞ্জাম নির্বাচন করার জন্য সংস্থাগুলিকে ক্ষমতা দেয়

Kostenloser Workshop von Security Risk Advisors unterstützt Unternehmen bei der Auswahl optimaler OT-Sicherheitstools

El taller gratuito de Security Risk Advisors permite a las organizaciones seleccionar herramientas de seguridad OT óptimas

Un atelier gratuit organisé par des conseillers en risques de sécurité permet aux organisations de sélectionner les outils de sécurité OT optimaux

सुरक्षा जोखिम सलाहकारों की ओर से निःशुल्क कार्यशाला संगठनों को इष्टतम OT सुरक्षा उपकरण चुनने में सक्षम बनाती है

セキュリティリスクアドバイザーによる無料ワークショップで、組織は最適な OT セキュリティツールを選択できるようになります。

보안 위험 자문가의 무료 워크숍을 통해 조직이 최적의 OT 보안 도구를 선택할 수 있도록 지원

Workshop gratuito de consultores de risco de segurança capacita organizações a selecionar ferramentas de segurança TO ideais

Бесплатный семинар от консультантов по рискам безопасности дает организациям возможность выбирать оптимальные инструменты безопасности OT

Güvenlik Riski Danışmanlarından Ücretsiz Atölye Çalışması Kuruluşlara En İyi OT Güvenlik Araçlarını Seçme Gücü Veriyor

Hội thảo miễn phí từ các cố vấn rủi ro bảo mật trao quyền cho các tổ chức lựa chọn các công cụ bảo mật OT tối ưu

安全风险顾问举办的免费研讨会帮助组织选择最佳 OT 安全工具

In these intensely competitive times, every application needs to be aesthetically appealing in addition to providing seamless functionality to satisfy the users. If an app does not function properly because of a visual interface error or if the site is unreadable, users may uninstall or leave the app. This can ultimately lead to a loss of customers and a drop in revenues. To avoid this and ensure the best possible user experience, companies are prioritizing visual regression testing. Let us study a couple of examples of visual regression testing to understand its importance: When you click on a drop-down image the menu does not slide down after adding more options.
You have reworked the complete layout of the page, and the submit button on the page is broken. These UI errors need to be fixed since changes in the elements can lead to the breakdown of the visual aesthetics. These illustrations clearly highlight the importance of conducting thorough visual regression testing before software releases. Visual regression testing checks the UI of an application after code changes and ensures that these different visual elements, like buttons, images, and text, all render correctly. Through this blog, we will examine visual regression, the different techniques used, the challenges testers face when performing the tests, and the best practices to overcome them. Common Visual Regression Techniques Visual testing ensures that visual glitches are identified early and do not spill over to the production stage. Let us now briefly examine the commonly used methods for regression testing: Manual visual testing: In this method, testers manually compare UI screenshots before and after making changes in the code.
Comparing layouts: It requires comparing the UI layout before and after changes are made and accounts for changes in the UI’s font, color, and other visual elements.
Comparing pixel-by-pixel: Used to test small, complex UI elements, this method involves comparing the UI’s pixels before and after changes.
Structural comparison: It compares the UI’s Document Object Model (DOM) before and after making changes.
Inserting visual checkpoints: In this technique, checkpoints are inserted into the UI to check the UI’s visual state at specific points in time. Manual Regression Testing Manual regression testing is recommended in the early testing stages when the UI is unstable. It is suitable for ad hoc testing and conducting quick on-the-spot checks. In this method, the testers manually check the layout and appearance of the visual elements. It involves taking the latest screenshots and comparing them with baseline screenshots to unearth any gaps. Benefits of Automated Visual Regression Testing Over Manual Methods The manual technique can be time-consuming and lead to a waste of resources because of its repetitive nature. The scale of manual testing grows with every new feature addition, thereby increasing the time spent by a human tester pouring over screenshots to identify bugs. It is impractical to manually conduct visual regression testing when there are tight deadlines to meet. In contrast, automated visual regression testing helps you automate repetitive tasks like checking the page following every code change. Even with the heavy initial investment, automated testing, especially with AI-powered features, helps you save time and money in the long run. Though automated visual regression testing reaps many benefits for testers and developers, it has to be properly implemented if they are to receive its benefits. Let us examine the best practices that go into a proper visual regression testing process. What To Remember While Conducting Visual Regression Testing Following these best practices will help optimize your automated visual regression testing process and help you gain its maximum benefits: Develop Appropriate Test Scenarios Clearly define what to record in the snapshots and the time to take them while testing. Remember to include user interactions in such scenarios since the applications will need to deal with them in the real world. Validate the Entire UI Page Instead of only validating individual components, ensure to validate the entire UI page. This practice will increase the test coverage and improve it. Pick the Right Automated Tool The visual regression testing tool you choose should be capable of ignoring false positives. It should be able to take care of pixel offsets, anti-aliasing, and other issues while ensuring the tests don’t fail. The automated algorithm must be able to analyze the page’s structure and perform layout comparisons. It should also be able to take care of content that is dynamic and moving. Compare Screenshots Testers should compare screenshots with recent ones to identify the disparities. This will help ensure that the UI remains the same, improving the user experience and product quality. Identify and Resolve Bugs Ensure to resolve bugs that the testing tool identifies or send them to the respective developers for fixing them. Remember to update screenshots to be used as a baseline for future regression tests after it is done. Choose the Right Situation To Perform Automated Testing Automated visual regression testing may not be appropriate for all scenarios. In some situations, using a combination of manual and automated methods is advisable. For example, when new features are added, they can be validated using an automation technique and then checked manually. Conduct a Detailed Review A team of reviewers or a single reviewer can examine how adjustments are made without causing interruptions in the desired outcome and make a report accordingly. Learn From Your Mistakes Learning from your successes and failures is crucial in visual regression testing. You need to identify the root causes and impacts of the visual changes detected by your tool. Testers must also identify false negatives and positives and adjust the tool settings and test cases accordingly. Conclusion In conclusion, successful implementation of visual regression testing in Agile teams requires a holistic approach that encompasses automation, collaboration, scalability, and comprehensive analysis. Visual regression testing can be a game-changer if you choose an appropriate automated testing tool with the right features and advantages. By adhering to these best practices, Agile teams can effectively leverage visual regression testing to deliver high-quality user interfaces in an iterative and efficient manner. In these intensely competitive times, every application needs to be aesthetically appealing in addition to providing seamless functionality to satisfy the users. If an app does not function properly because of a visual interface error or if the site is unreadable, users may uninstall or leave the app. This can ultimately lead to a loss of customers and a drop in revenues. To avoid this and ensure the best possible user experience, companies are prioritizing visual regression testing . visual regression testing Let us study a couple of examples of visual regression testing to understand its importance: Let us study a couple of examples of visual regression testing to understand its importance: When you click on a drop-down image the menu does not slide down after adding more options. You have reworked the complete layout of the page, and the submit button on the page is broken. When you click on a drop-down image the menu does not slide down after adding more options. You have reworked the complete layout of the page, and the submit button on the page is broken. These UI errors need to be fixed since changes in the elements can lead to the breakdown of the visual aesthetics. These illustrations clearly highlight the importance of conducting thorough visual regression testing before software releases. Visual regression testing checks the UI of an application after code changes and ensures that these different visual elements, like buttons, images, and text, all render correctly. Through this blog, we will examine visual regression, the different techniques used, the challenges testers face when performing the tests, and the best practices to overcome them. Common Visual Regression Techniques Visual testing ensures that visual glitches are identified early and do not spill over to the production stage. Let us now briefly examine the commonly used methods for regression testing: Manual visual testing: In this method, testers manually compare UI screenshots before and after making changes in the code. Comparing layouts: It requires comparing the UI layout before and after changes are made and accounts for changes in the UI’s font, color, and other visual elements. Comparing pixel-by-pixel: Used to test small, complex UI elements, this method involves comparing the UI’s pixels before and after changes. Structural comparison: It compares the UI’s Document Object Model (DOM) before and after making changes. Inserting visual checkpoints: In this technique, checkpoints are inserted into the UI to check the UI’s visual state at specific points in time. Manual visual testing: In this method, testers manually compare UI screenshots before and after making changes in the code. Manual visual testing: Comparing layouts: It requires comparing the UI layout before and after changes are made and accounts for changes in the UI’s font, color, and other visual elements. Comparing layouts: Comparing pixel-by-pixel: Used to test small, complex UI elements, this method involves comparing the UI’s pixels before and after changes. Comparing pixel-by-pixel: Structural comparison: It compares the UI’s Document Object Model (DOM) before and after making changes. Structural comparison: Inserting visual checkpoints: In this technique, checkpoints are inserted into the UI to check the UI’s visual state at specific points in time. Inserting visual checkpoints: Manual Regression Testing Manual regression testing is recommended in the early testing stages when the UI is unstable. It is suitable for ad hoc testing and conducting quick on-the-spot checks. In this method, the testers manually check the layout and appearance of the visual elements. It involves taking the latest screenshots and comparing them with baseline screenshots to unearth any gaps. Benefits of Automated Visual Regression Testing Over Manual Methods The manual technique can be time-consuming and lead to a waste of resources because of its repetitive nature. The scale of manual testing grows with every new feature addition, thereby increasing the time spent by a human tester pouring over screenshots to identify bugs. It is impractical to manually conduct visual regression testing when there are tight deadlines to meet. In contrast, automated visual regression testing helps you automate repetitive tasks like checking the page following every code change. Even with the heavy initial investment, automated testing, especially with AI-powered features, helps you save time and money in the long run. Though automated visual regression testing reaps many benefits for testers and developers, it has to be properly implemented if they are to receive its benefits. Let us examine the best practices that go into a proper visual regression testing process. What To Remember While Conducting Visual Regression Testing Following these best practices will help optimize your automated visual regression testing process and help you gain its maximum benefits: Develop Appropriate Test Scenarios Clearly define what to record in the snapshots and the time to take them while testing. Remember to include user interactions in such scenarios since the applications will need to deal with them in the real world. Validate the Entire UI Page Instead of only validating individual components, ensure to validate the entire UI page. This practice will increase the test coverage and improve it. Pick the Right Automated Tool The visual regression testing tool you choose should be capable of ignoring false positives. It should be able to take care of pixel offsets, anti-aliasing, and other issues while ensuring the tests don’t fail. The automated algorithm must be able to analyze the page’s structure and perform layout comparisons. It should also be able to take care of content that is dynamic and moving. Compare Screenshots Testers should compare screenshots with recent ones to identify the disparities. This will help ensure that the UI remains the same, improving the user experience and product quality. Identify and Resolve Bugs Ensure to resolve bugs that the testing tool identifies or send them to the respective developers for fixing them. Remember to update screenshots to be used as a baseline for future regression tests after it is done. Choose the Right Situation To Perform Automated Testing Automated visual regression testing may not be appropriate for all scenarios. In some situations, using a combination of manual and automated methods is advisable. For example, when new features are added, they can be validated using an automation technique and then checked manually. Conduct a Detailed Review A team of reviewers or a single reviewer can examine how adjustments are made without causing interruptions in the desired outcome and make a report accordingly. Learn From Your Mistakes Learning from your successes and failures is crucial in visual regression testing. You need to identify the root causes and impacts of the visual changes detected by your tool. Testers must also identify false negatives and positives and adjust the tool settings and test cases accordingly. Conclusion In conclusion, successful implementation of visual regression testing in Agile teams requires a holistic approach that encompasses automation, collaboration, scalability, and comprehensive analysis. Visual regression testing can be a game-changer if you choose an appropriate automated testing tool with the right features and advantages. By adhering to these best practices, Agile teams can effectively leverage visual regression testing to deliver high-quality user interfaces in an iterative and efficient manner.

Optimal Strategies for Integrating Visual Regression Testing into Agile Teams

Efficient Neural Network Approaches for Conditional Optimal Transport: Discussion and Reference

Efficient Neural Network Approaches for Conditional Optimal Transport: Numerical Experiments

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu;
(2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu;
(3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu;
(4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu;
(5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. Table of Links Abstract & Introduction
Background and Related Work
Partially Convex Potential Maps (PCP-Map) for Conditional OT
Conditional OT flow (COT-Flow)
Implementation and Experimental Setup
Numerical Experiments
Discussion and Reference 4. Conditional OT flow (COT-Flow). In this section, we extend the OT-regularized continuous normalizing flows in [36] to the conditional generative modeling problem introduced in section 2. Training Problem. Following the general approach of continuous normalizing flows [11, 19], we express the transport map that pushes forward the reference distribution ρz to the target π(x|y) via the flow map of an ODE. That is, we define g(z, y) = u(1) as the terminal state of the ODE where the evolution of u: [0, 1] → R n depends on the velocity field v : [0, 1] × R n × R m → R n and will be parameterized with a neural network below. When the velocity is trained to minimize the negative log-likelihood loss in (3.2), the resulting mapping is called a continuous normalizing flow [19]. We add y as an additional input to the velocity field to enable conditional sampling. Consequently, the resulting flow map and generator g depend on y. One advantage of defining the generator through an ODE is that the loss function can be evaluated efficiently for a wide range of velocity functions. Recall that the loss function requires the inverse of the generator and the log-determinant of its Jacobian. For sufficiently regular velocity fields, the inverse of the generator can be obtained by integrating backward in time. To be precise, we define g −1 (x, y) = p(0) where p : [0, 1] → R n satisfies (4.1) with the terminal condition p(1) = x. As derived in [58, 19, 56], constructing the generator through an ODE also simplifies computing the log-determinant of the Jacobian, i.e., Penalizing transport costs during training leads to theoretical and numerical advantages; see, for example, [58, 56, 36, 17]. Hence, we consider the OT-regularized training problem where α1 > 0 is a regularization parameter that trades off matching the distributions (for α1 ≪ 1) and minimizing the transport costs (for α1 ≫ 0) given by the dynamic transport cost penalty This penalty is stronger than the static counterpart in (3.4), in other words However, the values of both penalties agree when the velocity field in (4.1) is constant along the trajectories, which is the case for the optimal transport map. Hence, we expect the solution of the dynamic problem to be close to that of the static formulation when α1 is chosen well. To provide additional theoretical insight and motivate our numerical method, we note that (4.3) is related to the potential mean field game (MFG) Here, the terminal and running costs in the objective functional are the L2 anti-derivatives of JNLL and PDOT, respectively, and the continuity equation is used to represent the density evolution under the velocity v. To be precise, (4.3) can be seen as a discretization of (4.6) in the Lagrangian coordinates defined by the reversed ODE (4.14). Note that both formulations differ in the way the transport costs are measured: (4.4) computes the cost of pushing the conditional distribution to the Gaussian, while (4.6) penalizes the costs of pushing the Gaussian to the conditional. While these two terms do not generally agree, they coincide for the L2 optimal transport map; see [16, Corollary 2.5.13] for a proof for unconditional optimal transport. For more insights into MFGs and their relation to optimal transport and generative modeling, we refer to our prior work [43] and the more recent work [57, Sec 3.4]. The fact that the solutions of the microscopic version in (4.3) and the macroscopic version in (4.6) agree is remarkable and was first shown in the seminal work [28]. By Pontryagin’s maximum principle, the solution of (4.3) satisfies the feedback form where Φ : [0, 1] × R n × R m → R and Φ(·, ·, y) can be seen as the value function of the MFG and, alternatively, as the Lagrange multiplier in (4.6) for a fixed y. Therefore, we model the velocity as a conservative vector field as also proposed in [36, 43]. This also simplifies the computation of the log-determinant since the Jacobian of v is symmetric, and we note that Another consequence of optimal control theory is that the value function satisfies the Hamilton Jacobi Bellman (HJB) equations with the terminal condition which is only tractable if the joint density, π(x, y), is available. These n-dimensional PDEs are parameterized by y ∈ R m. Neural network approach. We parameterize the value function, Φ, with a scalar-valued neural network. In contrast to the static approach in the previous section, the choice of function approximator is more flexible in the dynamic approach. The approach can be effective as long as Φ is parameterized with any function approximation tool that is effective in high dimensions, allows efficient evaluation of its gradient and Laplacian, and the training problem can be solved sufficiently well. For our numerical experiments, we use the architecture considered in [36] and model Φ as the sum of a simple feed-forward neural network and a quadratic term. That is, As in [36], we model the neural network as a two-layer residual network of width w that reads Sample generation. After training, we draw i.i.d. samples from the approximated conditional distribution for a given y by sampling z ∼ ρz and solving the ODE (4.1). Since we train the inverse generator using a fixed number of integration steps, nt, it is interesting to investigate how close our solution is to the continuous problem. One indicator is to compare the samples obtained with different number of integration steps during sampling. Another indicator is to compute the variance of ∇Φ along the trajectories. Hyperparameters. As described in section 5, we randomly sample the width, w, of the neural network, the number of time integration steps during training, the penalty parameters α1, α2, and the hyperparameters of the Adam optimizer (batch size and initial learning rate) from the values listed in Table 2. This paper is available on arxiv under CC 4.0 license. Authors: (1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu; (2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu; (3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu; (4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu; (5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu; (2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu; (3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu; (4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu; (5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. Table of Links Abstract & Introduction Background and Related Work Partially Convex Potential Maps (PCP-Map) for Conditional OT Conditional OT flow (COT-Flow) Implementation and Experimental Setup Numerical Experiments Discussion and Reference Abstract & Introduction Abstract & Introduction Background and Related Work Background and Related Work Partially Convex Potential Maps (PCP-Map) for Conditional OT Partially Convex Potential Maps (PCP-Map) for Conditional OT Conditional OT flow (COT-Flow) Conditional OT flow (COT-Flow) Implementation and Experimental Setup Implementation and Experimental Setup Numerical Experiments Numerical Experiments Discussion and Reference Discussion and Reference 4. Conditional OT flow (COT-Flow). In this section, we extend the OT-regularized continuous normalizing flows in [36] to the conditional generative modeling problem introduced in section 2. Training Problem. Following the general approach of continuous normalizing flows [11, 19], we express the transport map that pushes forward the reference distribution ρz to the target π(x|y) via the flow map of an ODE. That is, we define g(z, y) = u(1) as the terminal state of the ODE where the evolution of u: [0, 1] → R n depends on the velocity field v : [0, 1] × R n × R m → R n and will be parameterized with a neural network below. When the velocity is trained to minimize the negative log-likelihood loss in (3.2), the resulting mapping is called a continuous normalizing flow [19]. We add y as an additional input to the velocity field to enable conditional sampling. Consequently, the resulting flow map and generator g depend on y. One advantage of defining the generator through an ODE is that the loss function can be evaluated efficiently for a wide range of velocity functions. Recall that the loss function requires the inverse of the generator and the log-determinant of its Jacobian. For sufficiently regular velocity fields, the inverse of the generator can be obtained by integrating backward in time. To be precise, we define g −1 (x, y) = p(0) where p : [0, 1] → R n satisfies (4.1) with the terminal condition p(1) = x. As derived in [58, 19, 56], constructing the generator through an ODE also simplifies computing the log-determinant of the Jacobian, i.e., Penalizing transport costs during training leads to theoretical and numerical advantages; see, for example, [58, 56, 36, 17]. Hence, we consider the OT-regularized training problem where α1 > 0 is a regularization parameter that trades off matching the distributions (for α1 ≪ 1) and minimizing the transport costs (for α1 ≫ 0) given by the dynamic transport cost penalty This penalty is stronger than the static counterpart in (3.4), in other words However, the values of both penalties agree when the velocity field in (4.1) is constant along the trajectories, which is the case for the optimal transport map. Hence, we expect the solution of the dynamic problem to be close to that of the static formulation when α1 is chosen well. To provide additional theoretical insight and motivate our numerical method, we note that (4.3) is related to the potential mean field game (MFG) Here, the terminal and running costs in the objective functional are the L2 anti-derivatives of JNLL and PDOT, respectively, and the continuity equation is used to represent the density evolution under the velocity v. To be precise, (4.3) can be seen as a discretization of (4.6) in the Lagrangian coordinates defined by the reversed ODE (4.14). Note that both formulations differ in the way the transport costs are measured: (4.4) computes the cost of pushing the conditional distribution to the Gaussian, while (4.6) penalizes the costs of pushing the Gaussian to the conditional. While these two terms do not generally agree, they coincide for the L2 optimal transport map; see [16, Corollary 2.5.13] for a proof for unconditional optimal transport. For more insights into MFGs and their relation to optimal transport and generative modeling, we refer to our prior work [43] and the more recent work [57, Sec 3.4]. The fact that the solutions of the microscopic version in (4.3) and the macroscopic version in (4.6) agree is remarkable and was first shown in the seminal work [28]. By Pontryagin’s maximum principle, the solution of (4.3) satisfies the feedback form where Φ : [0, 1] × R n × R m → R and Φ(·, ·, y) can be seen as the value function of the MFG and, alternatively, as the Lagrange multiplier in (4.6) for a fixed y. Therefore, we model the velocity as a conservative vector field as also proposed in [36, 43]. This also simplifies the computation of the log-determinant since the Jacobian of v is symmetric, and we note that Another consequence of optimal control theory is that the value function satisfies the Hamilton Jacobi Bellman (HJB) equations with the terminal condition which is only tractable if the joint density, π(x, y), is available. These n-dimensional PDEs are parameterized by y ∈ R m. Neural network approach. We parameterize the value function, Φ, with a scalar-valued neural network. In contrast to the static approach in the previous section, the choice of function approximator is more flexible in the dynamic approach. The approach can be effective as long as Φ is parameterized with any function approximation tool that is effective in high dimensions, allows efficient evaluation of its gradient and Laplacian, and the training problem can be solved sufficiently well. For our numerical experiments, we use the architecture considered in [36] and model Φ as the sum of a simple feed-forward neural network and a quadratic term. That is, As in [36], we model the neural network as a two-layer residual network of width w that reads Sample generation. After training, we draw i.i.d. samples from the approximated conditional distribution for a given y by sampling z ∼ ρz and solving the ODE (4.1). Since we train the inverse generator using a fixed number of integration steps, nt, it is interesting to investigate how close our solution is to the continuous problem. One indicator is to compare the samples obtained with different number of integration steps during sampling. Another indicator is to compute the variance of ∇Φ along the trajectories. Hyperparameters. As described in section 5, we randomly sample the width, w, of the neural network, the number of time integration steps during training, the penalty parameters α1, α2, and the hyperparameters of the Adam optimizer (batch size and initial learning rate) from the values listed in Table 2.

Efficient Neural Network Approaches for Conditional Optimal Transport:Conditional OT flow (COT-Flow)

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu;
(2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu;
(3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu;
(4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu;
(5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. Table of Links Abstract & Introduction
Background and Related Work
Partially Convex Potential Maps (PCP-Map) for Conditional OT
Conditional OT flow (COT-Flow)
Implementation and Experimental Setup
Numerical Experiments
Discussion and Reference Since there are usually infinitely many diffeomorphisms that satisfy (2.1), imposing additional properties such as monotonicity can effectively limit the search. Optimal transport theory can further motivate the restriction to monotone transport maps. Conditional optimal transport maps ensure equality in (2.1) while minimizing a specific transport cost. These maps often have additional structural properties that can be exploited during learning, leading to several promising works. For example, when using the L2 transport costs, a unique solution to the OT problem exists and is indeed a monotone operator; see [10] for theoretical results on the conditional OT problem. As observed in related approaches for (unconditional) deep generative modeling [36, 56, 58, 17], adding a transport cost penalty to the maximum likelihood training objective does not limit the ability to match the distributions. Measure transport approaches that introduce monotonicity through optimal transport theory include the CondOT approach [9], normalizing flows [20, 36, 58], and a GAN approach [3]. A more general framework for constructing OT-regularized transport maps is proposed by [56]. Many other measure transport approaches do not consider transport cost but enforce monotonicity by incorporating specific structures in their map parameterizations; see, for example [14, 26, 21, 39, 13, 15]. More generally, UMNN [53] provides a framework for constructing monotone neural networks. Other methods like the adaptive transport map (ATM) algorithm [4] enforce monotonicity through rectifier operators. There are some close relatives among the family of conditional optimal transport maps to our proposed approaches. The PCP-Map approach is most similar to the CP-Flow approach in [20]. CP-Flow is primarily designed to approximately solve the joint optimal transport problem. Its variant, however, allows for variational inference in the setting of a variational autoencoder (VAE). Inside the GitHub repository associated with [20], a script enables CP-Flow to approximately solve the COT problem over a 1D Gaussian mixture target distribution in a similar fashion to PCP-Map. Therefore, we compare our approach to the amortized CP-Flow using numerical experiments and find that the amortized CP-Flow either fails to solve the problems or is considerably slower than PCP-Map; see subsection 6.4. Our implementation of PCP-Map differs in the following aspects: a simplified transport map architecture, new automatic differentiation tools that avoid the need for stochastic log-determinant estimators, and a projected gradient method to enforce non-negativity constraints on parts of the weights. Another close relative is the CondOT approach in [9]. The main distinction between CondOT and PCP-Map is the definition of the learning problem. The former solves the W2 COT problem in an adversarial manner, which leads to a challenging stochastic saddle point problem. Our approach minimizes the L2 transport costs, which results in a minimization problem. The COT-Flow approach extends the dynamic OT formulation in [36] to enable conditional sampling and density estimation. This paper is available on arxiv under CC 4.0 license. Authors: (1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu; (2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu; (3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu; (4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu; (5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu; (2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu; (3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu; (4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu; (5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. Table of Links Abstract & Introduction Background and Related Work Partially Convex Potential Maps (PCP-Map) for Conditional OT Conditional OT flow (COT-Flow) Implementation and Experimental Setup Numerical Experiments Discussion and Reference Abstract & Introduction Abstract & Introduction Background and Related Work Background and Related Work Partially Convex Potential Maps (PCP-Map) for Conditional OT Partially Convex Potential Maps (PCP-Map) for Conditional OT Conditional OT flow (COT-Flow) Conditional OT flow (COT-Flow) Implementation and Experimental Setup Implementation and Experimental Setup Numerical Experiments Numerical Experiments Discussion and Reference Discussion and Reference Since there are usually infinitely many diffeomorphisms that satisfy (2.1), imposing additional properties such as monotonicity can effectively limit the search. Optimal transport theory can further motivate the restriction to monotone transport maps. Conditional optimal transport maps ensure equality in (2.1) while minimizing a specific transport cost. These maps often have additional structural properties that can be exploited during learning, leading to several promising works. For example, when using the L2 transport costs, a unique solution to the OT problem exists and is indeed a monotone operator; see [10] for theoretical results on the conditional OT problem. As observed in related approaches for (unconditional) deep generative modeling [36, 56, 58, 17], adding a transport cost penalty to the maximum likelihood training objective does not limit the ability to match the distributions. Measure transport approaches that introduce monotonicity through optimal transport theory include the CondOT approach [9], normalizing flows [20, 36, 58], and a GAN approach [3]. A more general framework for constructing OT-regularized transport maps is proposed by [56]. Many other measure transport approaches do not consider transport cost but enforce monotonicity by incorporating specific structures in their map parameterizations; see, for example [14, 26, 21, 39, 13, 15]. More generally, UMNN [53] provides a framework for constructing monotone neural networks. Other methods like the adaptive transport map (ATM) algorithm [4] enforce monotonicity through rectifier operators. There are some close relatives among the family of conditional optimal transport maps to our proposed approaches. The PCP-Map approach is most similar to the CP-Flow approach in [20]. CP-Flow is primarily designed to approximately solve the joint optimal transport problem. Its variant, however, allows for variational inference in the setting of a variational autoencoder (VAE). Inside the GitHub repository associated with [20], a script enables CP-Flow to approximately solve the COT problem over a 1D Gaussian mixture target distribution in a similar fashion to PCP-Map. Therefore, we compare our approach to the amortized CP-Flow using numerical experiments and find that the amortized CP-Flow either fails to solve the problems or is considerably slower than PCP-Map; see subsection 6.4. Our implementation of PCP-Map differs in the following aspects: a simplified transport map architecture, new automatic differentiation tools that avoid the need for stochastic log-determinant estimators, and a projected gradient method to enforce non-negativity constraints on parts of the weights. Another close relative is the CondOT approach in [9]. The main distinction between CondOT and PCP-Map is the definition of the learning problem. The former solves the W2 COT problem in an adversarial manner, which leads to a challenging stochastic saddle point problem. Our approach minimizes the L2 transport costs, which results in a minimization problem. The COT-Flow approach extends the dynamic OT formulation in [36] to enable conditional sampling and density estimation.

Efficient Neural Network Approaches for Conditional Optimal Transport: Background and Related Work

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu;
(2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu;
(3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu;
(4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu;
(5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. Table of Links Abstract & Introduction
Background and Related Work
Partially Convex Potential Maps (PCP-Map) for Conditional OT
Conditional OT flow (COT-Flow)
Implementation and Experimental Setup
Numerical Experiments
Discussion and ReferenceBayesianInference Abstract. We present two neural network approaches that approximate the solutions of static and dynamic conditional optimal transport (COT) problems, respectively. Both approaches enable sampling and density estimation of conditional probability distributions, which are core tasks in Bayesian inference. Our methods represent the target conditional distributions as transformations of a tractable reference distribution and, therefore, fall into the framework of measure transport. COT maps are a canonical choice within this framework, with desirable properties such as uniqueness and monotonicity. However, the associated COT problems are computationally challenging, even in moderate dimensions. To improve the scalability, our numerical algorithms leverage neural networks to parameterize COT maps. Our methods exploit the structure of the static and dynamic formulations of the COT problem. PCP-Map models conditional transport maps as the gradient of a partially input convex neural network (PICNN) and uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. COT-Flow models conditional transports via the flow of a regularized neural ODE; it is slower to train but offers faster sampling. We demonstrate their effectiveness and efficiency by comparing them with state-of-the art approaches using benchmark datasets and Bayesian inverse problems. Key words. Measure transport, generative modeling, optimal transport, Bayesian inference, inverse problems, uncertainty quantification MSC codes. 62F15, 62M45 Most effective sampling techniques are limited to the conventional Bayesian setting as they require a tractable likelihood model (often given by a forward operator that maps x to y and a noise model) and prior. Even in the conventional setting, producing thousands of approximately i.i.d. samples from the conditional distribution often requires millions of forward operator evaluations. This can be prohibitive for complex forward operators, such as those in science and engineering applications based on stochastic differential equations (SDE) or partial differential equations (PDE). Moreover, MCMC schemes are sequential and difficult to parallelize. Beyond the conventional Bayesian setting, one is usually limited to likelihood-free methods; see, e.g., [12]. Among these methods, measure transport provides a general framework to characterize complex posteriors using samples from the joint distribution. The key idea is to construct transport maps that push forward a simple reference (e.g., a standard Gaussian) toward a complex target distribution; see [3] for discussions and reviews. Once obtained, these transport maps provide an immediate way to generate i.i.d. samples from the target distribution by evaluating the map at samples from a reference distribution. Under mild assumptions, there exist infinitely many transport maps that fulfill the push-forward constraint but have drastically different theoretical properties. One way to establish uniqueness is to identify the transport map that satisfies the push-forward constraint and incurs minimal transport cost. Adding transport costs renders the measure transport problem for the conditional distribution into a conditional optimal transport (COT) problem [52]. Solving COT problems is computationally challenging, especially when the number of parameters, n, or the number of measurements, m, are large or infinite. The curse of dimensionality affects methods that use grids or polynomials to approximate transport maps and renders most of them impractical when n + m is larger than ten. Due to their function approximation properties, neural networks are a natural candidate to parameterize COT maps. This choice bridges the measure transport framework and deep generative modeling [38, 27]. While showing promising results, many recent neural network approaches for COT such as [32, 9, 3] rely on adversarial training, which requires solving a challenging stochastic saddle point problem. This paper contributes two neural network approaches for COT that can be trained by maximizing the likelihood of the target samples, and we demonstrate their use in Bayesian inference. Our approaches exploit the known structure of the COT map in different ways. Our first approach parameterizes the map as the gradient of a PICNN [2]; we name it partially convex potential map (PCP-Map). By construction, this yields a monotone map for any choice of network weights. When trained to sufficient accuracy, we obtain the optimal transport map with respect to an expected L2 cost, known as the conditional Brenier map [10]. Our second approach builds upon the relaxed dynamical formulation of the L2 optimal transport problem that instead seeks a map defined by the flow map of an ODE; we name it conditional optimal transport flow (COT-Flow). Here, we parameterize the velocity of the map as the gradient of a scalar potential and obtain a neural ODE. To ensure that the network achieves sufficient accuracy, we monitor and penalize violations of the associated optimality conditions, which are given by a Hamilton Jacobi Bellman (HJB) PDE. A series of numerical experiments are conducted to evaluate PCP-Map and COT-Flow comprehensively. The first experiment demonstrates our approaches’ robustness to hyperparameters and superior numerical accuracy for density estimation compared to results from other approaches in [4] using six UCI tabular datasets [29]. The second experiment demonstrates our approaches’ effectiveness and efficiency by comparing them to a provably convergent approximate Bayesian computation approach on the task of conditional sampling using a Bayesian inference problem involving the stochastic Lotka-Volterra equation, which gives rise to intractable likelihood. The third experiment compares our approaches’ competitiveness against the flow-based neural posterior estimation (NPE) approach studied in [40] on a real-world high-dimensional Bayesian inference problem involving the 1D shallow water equations. The final experiment demonstrates PCP-Map’s improvements in computational stability and efficiency over an amortized version of the approach relevant to [20]. Through these experiments, we conclude that the proposed approaches characterize conditional distributions with improved numerical accuracy and efficiency. Moreover, they improve upon recent computational methods for the numerical solution of the COT problem. Like most neural network approaches, the effectiveness of our methods relies on an adequate choice of network architecture and an accurate solution to a stochastic non-convex optimization problem. As there is little theoretical guidance for choosing the network architecture and optimization hyper-parameters, our numerical experiments show the effectiveness and robustness of a simple random grid search. The choice of the optimization algorithm is a modular component in our approach; we use the common Adam method [24] for simplicity of implementation. The remainder of the paper is organized as follows: Section 2 contains the mathematical formulation of the conditional sampling problem and reviews related learning approaches. Section 3 presents our partially convex potential map (PCP-Map). Section 4 presents our conditional optimal transport flow (COT-Flow). Section 5 describes our effort to achieve reproducible results and procedures for identifying effective hyperparameters for neural network training. Section 6 contains a detailed numerical evaluation of both approaches using six open-source data sets and experiments motivated by Bayesian inference for the stochastic Lotka-Volterra equation and the 1D shallow water equation. Section 7 features a detailed discussion of our results and highlights the advantages and limitations of the presented approaches. This paper is available on arxiv under CC 4.0 license. Authors: (1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu; (2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu; (3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu; (4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu; (5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Zheyu Oliver Wang, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and olivrw@mit.edu; (2)  Ricardo Baptista, Computing + Mathematical Sciences, California Institute of Technology, Pasadena, CA and rsb@caltech.edu; (3) Youssef Marzouk, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA and ymarz@mit.edu; (4) Lars Ruthotto, Department of Mathematics, Emory University, Atlanta, GA and lruthotto@emory.edu; (5) Deepanshu Verma, Department of Mathematics, Emory University, Atlanta, GA and deepanshu.verma@emory.edu. Table of Links Abstract & Introduction Background and Related Work Partially Convex Potential Maps (PCP-Map) for Conditional OT Conditional OT flow (COT-Flow) Implementation and Experimental Setup Numerical Experiments Discussion and ReferenceBayesianInference Abstract & Introduction Abstract & Introduction Background and Related Work Background and Related Work Partially Convex Potential Maps (PCP-Map) for Conditional OT Partially Convex Potential Maps (PCP-Map) for Conditional OT Conditional OT flow (COT-Flow) Conditional OT flow (COT-Flow) Implementation and Experimental Setup Implementation and Experimental Setup Numerical Experiments Numerical Experiments Discussion and Reference BayesianInference Discussion and Reference Abstract. We present two neural network approaches that approximate the solutions of static and dynamic conditional optimal transport (COT) problems, respectively. Both approaches enable sampling and density estimation of conditional probability distributions, which are core tasks in Bayesian inference. Our methods represent the target conditional distributions as transformations of a tractable reference distribution and, therefore, fall into the framework of measure transport. COT maps are a canonical choice within this framework, with desirable properties such as uniqueness and monotonicity. However, the associated COT problems are computationally challenging, even in moderate dimensions. To improve the scalability, our numerical algorithms leverage neural networks to parameterize COT maps. Our methods exploit the structure of the static and dynamic formulations of the COT problem. PCP-Map models conditional transport maps as the gradient of a partially input convex neural network (PICNN) and uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. COT-Flow models conditional transports via the flow of a regularized neural ODE; it is slower to train but offers faster sampling. We demonstrate their effectiveness and efficiency by comparing them with state-of-the art approaches using benchmark datasets and Bayesian inverse problems. Key words. Measure transport, generative modeling, optimal transport, Bayesian inference, inverse problems, uncertainty quantification Key words. MSC codes. 62F15, 62M45 MSC codes. Most effective sampling techniques are limited to the conventional Bayesian setting as they require a tractable likelihood model (often given by a forward operator that maps x to y and a noise model) and prior. Even in the conventional setting, producing thousands of approximately i.i.d. samples from the conditional distribution often requires millions of forward operator evaluations. This can be prohibitive for complex forward operators, such as those in science and engineering applications based on stochastic differential equations (SDE) or partial differential equations (PDE). Moreover, MCMC schemes are sequential and difficult to parallelize. Beyond the conventional Bayesian setting, one is usually limited to likelihood-free methods; see, e.g., [12]. Among these methods, measure transport provides a general framework to characterize complex posteriors using samples from the joint distribution. The key idea is to construct transport maps that push forward a simple reference (e.g., a standard Gaussian) toward a complex target distribution; see [3] for discussions and reviews. Once obtained, these transport maps provide an immediate way to generate i.i.d. samples from the target distribution by evaluating the map at samples from a reference distribution. Under mild assumptions, there exist infinitely many transport maps that fulfill the push-forward constraint but have drastically different theoretical properties. One way to establish uniqueness is to identify the transport map that satisfies the push-forward constraint and incurs minimal transport cost. Adding transport costs renders the measure transport problem for the conditional distribution into a conditional optimal transport (COT) problem [52]. Solving COT problems is computationally challenging, especially when the number of parameters, n, or the number of measurements, m, are large or infinite. The curse of dimensionality affects methods that use grids or polynomials to approximate transport maps and renders most of them impractical when n + m is larger than ten. Due to their function approximation properties, neural networks are a natural candidate to parameterize COT maps. This choice bridges the measure transport framework and deep generative modeling [38, 27]. While showing promising results, many recent neural network approaches for COT such as [32, 9, 3] rely on adversarial training, which requires solving a challenging stochastic saddle point problem. This paper contributes two neural network approaches for COT that can be trained by maximizing the likelihood of the target samples, and we demonstrate their use in Bayesian inference. Our approaches exploit the known structure of the COT map in different ways. Our first approach parameterizes the map as the gradient of a PICNN [2]; we name it partially convex potential map (PCP-Map). By construction, this yields a monotone map for any choice of network weights. When trained to sufficient accuracy, we obtain the optimal transport map with respect to an expected L2 cost, known as the conditional Brenier map [10]. Our second approach builds upon the relaxed dynamical formulation of the L2 optimal transport problem that instead seeks a map defined by the flow map of an ODE; we name it conditional optimal transport flow (COT-Flow). Here, we parameterize the velocity of the map as the gradient of a scalar potential and obtain a neural ODE. To ensure that the network achieves sufficient accuracy, we monitor and penalize violations of the associated optimality conditions, which are given by a Hamilton Jacobi Bellman (HJB) PDE. A series of numerical experiments are conducted to evaluate PCP-Map and COT-Flow comprehensively. The first experiment demonstrates our approaches’ robustness to hyperparameters and superior numerical accuracy for density estimation compared to results from other approaches in [4] using six UCI tabular datasets [29]. The second experiment demonstrates our approaches’ effectiveness and efficiency by comparing them to a provably convergent approximate Bayesian computation approach on the task of conditional sampling using a Bayesian inference problem involving the stochastic Lotka-Volterra equation, which gives rise to intractable likelihood. The third experiment compares our approaches’ competitiveness against the flow-based neural posterior estimation (NPE) approach studied in [40] on a real-world high-dimensional Bayesian inference problem involving the 1D shallow water equations. The final experiment demonstrates PCP-Map’s improvements in computational stability and efficiency over an amortized version of the approach relevant to [20]. Through these experiments, we conclude that the proposed approaches characterize conditional distributions with improved numerical accuracy and efficiency. Moreover, they improve upon recent computational methods for the numerical solution of the COT problem. Like most neural network approaches, the effectiveness of our methods relies on an adequate choice of network architecture and an accurate solution to a stochastic non-convex optimization problem. As there is little theoretical guidance for choosing the network architecture and optimization hyper-parameters, our numerical experiments show the effectiveness and robustness of a simple random grid search. The choice of the optimization algorithm is a modular component in our approach; we use the common Adam method [24] for simplicity of implementation. The remainder of the paper is organized as follows: Section 2 contains the mathematical formulation of the conditional sampling problem and reviews related learning approaches. Section 3 presents our partially convex potential map (PCP-Map). Section 4 presents our conditional optimal transport flow (COT-Flow). Section 5 describes our effort to achieve reproducible results and procedures for identifying effective hyperparameters for neural network training. Section 6 contains a detailed numerical evaluation of both approaches using six open-source data sets and experiments motivated by Bayesian inference for the stochastic Lotka-Volterra equation and the 1D shallow water equation. Section 7 features a detailed discussion of our results and highlights the advantages and limitations of the presented approaches.

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