A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints

Roussel Desmond Nzoyem; David A W Barton; Tom Deakin

doi:10.1145/3624062.3626078

A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints

Roussel Desmond Nzoyem, David A W Barton, Tom Deakin

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

Abstract

The field of Optimal Control under Partial Differential Equations (PDE) constraints is rapidly changing under the influence of Deep Learning and the accompanying automatic differentiation libraries. Novel techniques like Physics-Informed Neural Networks (PINNs) and Differentiable Programming (DP) are to be contrasted with established numerical schemes like Direct-Adjoint Looping (DAL). We present a comprehensive comparison of DAL, PINN, and DP using a general-purpose mesh-free differentiable PDE solver based on Radial Basis Functions. Under Laplace and Navier-Stokes equations, we found DP to be extremely effective as it produces the most accurate gradients; thriving even when DAL fails and PINNs struggle. Additionally, we provide a detailed benchmark highlighting the limited conditions under which any of those methods can be efficiently used. Our work provides a guide to Optimal Control practitioners and connects them further to the Deep Learning community.

Original language	English
Title of host publication	SC-W '23: Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis
Editors	Dorian C. Arnold
Publisher	Association for Computing Machinery (ACM)
Pages	21-28
Number of pages	8
ISBN (Electronic)	9798400707858
DOIs	https://doi.org/10.1145/3624062.3626078
Publication status	Published - 12 Nov 2023
Event	Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis: Workshop on Artificial Intelligence and Machine Learning for Scientific Applications - Denver, United States Duration: 12 Nov 2023 → 17 Nov 2023 https://ai4s.github.io/

Workshop

Workshop	Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis
Abbreviated title	AI4S'23
Country/Territory	United States
City	Denver
Period	12/11/23 → 17/11/23
Internet address	https://ai4s.github.io/

Research Groups and Themes

Engineering Mathematics Research Group

Access to Document

10.1145/3624062.3626078Licence: CC BY

https://arxiv.org/abs/2310.02286Licence: Unspecified

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Nzoyem, R. D., Barton, D. A. W., & Deakin, T. (2023). A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints. In D. C. Arnold (Ed.), SC-W '23: Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis (pp. 21-28). Association for Computing Machinery (ACM). https://doi.org/10.1145/3624062.3626078

Nzoyem, Roussel Desmond ; Barton, David A W ; Deakin, Tom. / A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints. SC-W '23: Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis. editor / Dorian C. Arnold. Association for Computing Machinery (ACM), 2023. pp. 21-28

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abstract = "The field of Optimal Control under Partial Differential Equations (PDE) constraints is rapidly changing under the influence of Deep Learning and the accompanying automatic differentiation libraries. Novel techniques like Physics-Informed Neural Networks (PINNs) and Differentiable Programming (DP) are to be contrasted with established numerical schemes like Direct-Adjoint Looping (DAL). We present a comprehensive comparison of DAL, PINN, and DP using a general-purpose mesh-free differentiable PDE solver based on Radial Basis Functions. Under Laplace and Navier-Stokes equations, we found DP to be extremely effective as it produces the most accurate gradients; thriving even when DAL fails and PINNs struggle. Additionally, we provide a detailed benchmark highlighting the limited conditions under which any of those methods can be efficiently used. Our work provides a guide to Optimal Control practitioners and connects them further to the Deep Learning community.",

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Nzoyem, RD , Barton, DAW & Deakin, T 2023, A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints. in D C. Arnold (ed.), SC-W '23: Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis. Association for Computing Machinery (ACM), pp. 21-28, Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis, Denver, Colorado, United States, 12/11/23. https://doi.org/10.1145/3624062.3626078

A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints. / Nzoyem, Roussel Desmond ; Barton, David A W ; Deakin, Tom.
SC-W '23: Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis. ed. / Dorian C. Arnold. Association for Computing Machinery (ACM), 2023. p. 21-28.

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

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PY - 2023/11/12

Y1 - 2023/11/12

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AB - The field of Optimal Control under Partial Differential Equations (PDE) constraints is rapidly changing under the influence of Deep Learning and the accompanying automatic differentiation libraries. Novel techniques like Physics-Informed Neural Networks (PINNs) and Differentiable Programming (DP) are to be contrasted with established numerical schemes like Direct-Adjoint Looping (DAL). We present a comprehensive comparison of DAL, PINN, and DP using a general-purpose mesh-free differentiable PDE solver based on Radial Basis Functions. Under Laplace and Navier-Stokes equations, we found DP to be extremely effective as it produces the most accurate gradients; thriving even when DAL fails and PINNs struggle. Additionally, we provide a detailed benchmark highlighting the limited conditions under which any of those methods can be efficiently used. Our work provides a guide to Optimal Control practitioners and connects them further to the Deep Learning community.

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T2 - Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis

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Nzoyem RD , Barton DAW , Deakin T. A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints. In C. Arnold D, editor, SC-W '23: Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis. Association for Computing Machinery (ACM). 2023. p. 21-28 doi: 10.1145/3624062.3626078

A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints

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