Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems

Chen, Paula; Meng, Tingwei; Zou, Zongren; Darbon, Jérôme; Karniadakis, George Em

doi:10.1137/23m1561397

Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems

Journal Article · Fri Mar 15 00:00:00 EDT 2024 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/23m1561397· OSTI ID:2441157

Chen, Paula ^[1]; ^[2]; Zou, Zongren ^[3]; ^[3]; ^[4]

Brown University, Providence, RI (United States); Brown University
University of California, Los Angeles, CA (United States)
Brown University, Providence, RI (United States)
Brown University, Providence, RI (United States); Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Lastly, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.

View Accepted Manuscript (DOE)

Research Organization:: Brown University, Providence, RI (United States)

Sponsoring Organization:: Air Force Office of Scientific Research (AFOSR); Multidisciplinary University Research Initiative (MURI); USDOE Office of Science (SC)

Grant/Contract Number:: SC0023191

OSTI ID:: 2441157

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 2 Vol. 46; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (29)

Numerical Methods for Ordinary Differential Equations Butcher, J. C. https://doi.org/10.1002/9781119121534	book	July 2016
Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations Yegorov, Ivan; Dower, Peter M. Applied Mathematics & Optimization, Vol. 83, Issue 1 https://doi.org/10.1007/s00245-018-9509-6	journal	July 2018
A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging Chambolle, Antonin; Pock, Thomas Journal of Mathematical Imaging and Vision, Vol. 40, Issue 1 https://doi.org/10.1007/s10851-010-0251-1	journal	December 2010
On Hamilton–Jacobi PDEs and Image Denoising Models with Certain Nonadditive Noise Darbon, Jérôme; Meng, Tingwei; Resmerita, Elena Journal of Mathematical Imaging and Vision, Vol. 64, Issue 4 https://doi.org/10.1007/s10851-022-01073-3	journal	March 2022
Error propagation properties of recursive least-squares adaptation algorithms Ljung, Stefan; Ljung, Lennart Automatica, Vol. 21, Issue 2 https://doi.org/10.1016/0005-1098(85)90110-4	journal	March 1985
The taxation principle and multi-time Hamilton-Jacobi equations Rochet, J. C. Journal of Mathematical Economics, Vol. 14, Issue 2 https://doi.org/10.1016/0304-4068(85)90015-1	journal	January 1985
On Hopf's formulas for solutions of Hamilton-Jacobi equations Bardi, M.; Evans, L. C. Nonlinear Analysis: Theory, Methods & Applications, Vol. 8, Issue 11 https://doi.org/10.1016/0362-546X(84)90020-8	journal	January 1984
Discovering a reaction–diffusion model for Alzheimer’s disease by combining PINNs with symbolic regression Zhang, Zhen; Zou, Zongren; Kuhl, Ellen Computer Methods in Applied Mechanics and Engineering, Vol. 419 https://doi.org/10.1016/j.cma.2023.116647	journal	February 2024
DGM: A deep learning algorithm for solving partial differential equations Sirignano, Justin; Spiliopoulos, Konstantinos Journal of Computational Physics, Vol. 375 https://doi.org/10.1016/j.jcp.2018.08.029	journal	December 2018
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Raissi, M.; Perdikaris, P.; Karniadakis, G. E. Journal of Computational Physics, Vol. 378 https://doi.org/10.1016/j.jcp.2018.10.045	journal	February 2019
Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons Psaros, Apostolos F.; Meng, Xuhui; Zou, Zongren Journal of Computational Physics, Vol. 477 https://doi.org/10.1016/j.jcp.2022.111902	journal	March 2023
Continual lifelong learning with neural networks: A review Parisi, German I.; Kemker, Ronald; Part, Jose L. Neural Networks, Vol. 113 https://doi.org/10.1016/j.neunet.2019.01.012	journal	May 2019
Deep learning LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey Nature, Vol. 521, Issue 7553 https://doi.org/10.1038/nature14539	journal	May 2015
Array programming with NumPy Harris, Charles R.; Millman, K. Jarrod; van der Walt, Stéfan J. Nature, Vol. 585, Issue 7825 https://doi.org/10.1038/s41586-020-2649-2	journal	September 2020
Deep transfer operator learning for partial differential equations under conditional shift Goswami, Somdatta; Kontolati, Katiana; Shields, Michael D. Nature Machine Intelligence, Vol. 4, Issue 12, p. 1155-1164 https://doi.org/10.1038/s42256-022-00569-2	journal	December 2022
Discovering governing equations from data by sparse identification of nonlinear dynamical systems Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan Proceedings of the National Academy of Sciences, Vol. 113, Issue 15 https://doi.org/10.1073/pnas.1517384113	journal	March 2016
Overcoming catastrophic forgetting in neural networks Kirkpatrick, James; Pascanu, Razvan; Rabinowitz, Neil Proceedings of the National Academy of Sciences, Vol. 114, Issue 13 https://doi.org/10.1073/pnas.1611835114	journal	March 2017
Solving high-dimensional partial differential equations using deep learning Han, Jiequn; Jentzen, Arnulf; E., Weinan Proceedings of the National Academy of Sciences, Vol. 115, Issue 34 https://doi.org/10.1073/pnas.1718942115	journal	August 2018
Sampled limited memory methods for massive linear inverse problems Chung, Julianne; Chung, Matthias; Tanner Slagel, J. Inverse Problems, Vol. 36, Issue 5 https://doi.org/10.1088/1361-6420/ab77da	journal	April 2020
Hopf formula and multitime Hamilton-Jacobi equations Lions, P. -L.; Rochet, J. -C. Proceedings of the American Mathematical Society, Vol. 96, Issue 1 https://doi.org/10.1090/S0002-9939-1986-0813815-5	journal	January 1986
On the numerical stability and accuracy of the conventional recursive least squares algorithm Liavas, A. P.; Regalia, P. A. IEEE Transactions on Signal Processing, Vol. 47, Issue 1 https://doi.org/10.1109/78.738242	journal	January 1999
QRnet: Optimal Regulator Design With LQR-Augmented Neural Networks Nakamura-Zimmerer, Tenavi; Gong, Qi; Kang, Wei IEEE Control Systems Letters, Vol. 5, Issue 4 https://doi.org/10.1109/LCSYS.2020.3034415	journal	October 2021
Pareto-Based Multiobjective Machine Learning: An Overview and Case Studies No authors listed IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), Vol. 38, Issue 3 https://doi.org/10.1109/TSMCC.2008.919172	journal	May 2008
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures Wan, Xiaoliang; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 28, Issue 3 https://doi.org/10.1137/050627630	journal	January 2006
On Convex Finite-Dimensional Variational Methods in Imaging Sciences and Hamilton--Jacobi Equations Darbon, Jérôme SIAM Journal on Imaging Sciences, Vol. 8, Issue 4 https://doi.org/10.1137/130944163	journal	January 2015
On Decomposition Models in Imaging Sciences and Multi-time Hamilton--Jacobi Partial Differential Equations Darbon, Jérôme; Meng, Tingwei SIAM Journal on Imaging Sciences, Vol. 13, Issue 2 https://doi.org/10.1137/19M1266332	journal	January 2020
NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators Zou, Zongren; Meng, Xuhui; Psaros, Apostolos F. SIAM Review, Vol. 66, Issue 1 https://doi.org/10.1137/22M1518189	journal	February 2024
Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere Darbon, Jérôme; Osher, Stanley Research in the Mathematical Sciences, Vol. 3, Issue 1 https://doi.org/10.1186/s40687-016-0068-7	journal	September 2016
Advances and Open Problems in Federated Learning Kairouz, Peter; McMahan, H. Brendan; Avent, Brendan Foundations and Trends® in Machine Learning, Vol. 14, Issue 1–2 https://doi.org/10.1561/2200000083	journal	January 2021

Similar Records

Equivalence between Nonlinear H{sub {infinity}} Control Problems and Existence of Viscosity Solutions of Hamilton-Jacobi-Isaacs Equations

Journal Article · Thu Jan 14 23:00:00 EST 1999 · Applied Mathematics and Optimization · OSTI ID:21067556

Regularity and Variationality of Solutions to Hamilton-Jacobi Equations. Part II: Variationality, Existence, Uniqueness

Journal Article · Fri Apr 15 00:00:00 EDT 2011 · Applied Mathematics and Optimization · OSTI ID:22043935

Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients

Journal Article · Sat Dec 30 23:00:00 EST 1995 · Communications in Partial Differential Equations · OSTI ID:482485

Related Subjects

97 MATHEMATICS AND COMPUTING
Hopf formula
Riccati equation
linear quadratic regulator
linear regression
machine learning
multitime Hamilton-Jacobi PDEs

Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems

Citation Formats

References (29)

Similar Records

Related Subjects