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Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

Publication ,  Journal Article
Wang, M; Lu, J
Published in: Communications in Mathematics and Statistics
March 1, 2023

In this paper, we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L1 sense. The weak formulations are derived following (Brenier in Examples of hidden convexity in nonlinear PDEs, 2020) which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods, which we hope can provide some useful experience for future investigations.

Duke Scholars

Published In

Communications in Mathematics and Statistics

DOI

EISSN

2194-671X

ISSN

2194-6701

Publication Date

March 1, 2023

Volume

11

Issue

1

Start / End Page

21 / 57
 

Citation

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ICMJE
MLA
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Wang, M., & Lu, J. (2023). Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions. Communications in Mathematics and Statistics, 11(1), 21–57. https://doi.org/10.1007/s40304-023-00339-5
Wang, M., and J. Lu. “Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions.” Communications in Mathematics and Statistics 11, no. 1 (March 1, 2023): 21–57. https://doi.org/10.1007/s40304-023-00339-5.
Wang M, Lu J. Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions. Communications in Mathematics and Statistics. 2023 Mar 1;11(1):21–57.
Wang, M., and J. Lu. “Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions.” Communications in Mathematics and Statistics, vol. 11, no. 1, Mar. 2023, pp. 21–57. Scopus, doi:10.1007/s40304-023-00339-5.
Wang M, Lu J. Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions. Communications in Mathematics and Statistics. 2023 Mar 1;11(1):21–57.
Journal cover image

Published In

Communications in Mathematics and Statistics

DOI

EISSN

2194-671X

ISSN

2194-6701

Publication Date

March 1, 2023

Volume

11

Issue

1

Start / End Page

21 / 57