Solving partial differential equations with sampled neural networks
<|reference_start|>Solving partial differential equations with sampled neural networks: Approximation of solutions to partial differential equations (PDE) is an important problem in computational science and engineering. Using neural networks as an ansatz for the solution has proven a challenge in terms of training time and approximation accuracy. In this contribution, we discuss how sampling the hidden weights and biases of the ansatz network from data-agnostic and data-dependent probability distributions allows us to progress on both challenges. In most examples, the random sampling schemes outperform iterative, gradient-based optimization of physics-informed neural networks regarding training time and accuracy by several orders of magnitude. For time-dependent PDE, we construct neural basis functions only in the spatial domain and then solve the associated ordinary differential equation with classical methods from scientific computing over a long time horizon. This alleviates one of the greatest challenges for neural PDE solvers because it does not require us to parameterize the solution in time. For second-order elliptic PDE in Barron spaces, we prove the existence of sampled networks with $L^2$ convergence to the solution. We demonstrate our approach on several time-dependent and static PDEs. We also illustrate how sampled networks can effectively solve inverse problems in this setting. Benefits compared to common numerical schemes include spectral convergence and mesh-free construction of basis functions.<|reference_end|>
@article{datar2024solving,
    title={Solving partial differential equations with sampled neural networks},
    author={Chinmay Datar, Taniya Kapoor, Abhishek Chandra, Qing Sun, Iryna Burak,
  Erik Lien Bolager, Anna Veselovska, Massimo Fornasier, Felix Dietrich},
    journal={arXiv preprint arXiv:2405.20836},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.20836},
    primaryClass={math.NA cs.LG cs.NA}
}