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

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Abstract

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 \(L^1\) 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.

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Funding

This work is supported in part by National Science Foundation via grant DMS- 2012286 and by Department of Energy via grant DE-SC0019449.

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  1. Duke University, Durham, NC, USA

    Min Wang & Jianfeng Lu

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  1. Min Wang
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  2. Jianfeng Lu
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Correspondence to Min Wang.

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Wang, M., Lu, J. Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions. Commun. Math. Stat. 11, 21–57 (2023). https://doi.org/10.1007/s40304-023-00339-5

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  • DOI: https://doi.org/10.1007/s40304-023-00339-5

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