Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks
This addresses the problem of efficient and accurate neural network training for evolution equations, offering a domain-specific incremental improvement over existing methods.
The paper tackles the challenge of error accumulation in sequential-in-time training of neural networks for solving time-dependent partial differential equations by introducing Neural Galerkin schemes with randomized sparse updates, achieving up to two orders of magnitude improvement in accuracy and speed in numerical experiments.
Training neural networks sequentially in time to approximate solution fields of time-dependent partial differential equations can be beneficial for preserving causality and other physics properties; however, the sequential-in-time training is numerically challenging because training errors quickly accumulate and amplify over time. This work introduces Neural Galerkin schemes that update randomized sparse subsets of network parameters at each time step. The randomization avoids overfitting locally in time and so helps prevent the error from accumulating quickly over the sequential-in-time training, which is motivated by dropout that addresses a similar issue of overfitting due to neuron co-adaptation. The sparsity of the update reduces the computational costs of training without losing expressiveness because many of the network parameters are redundant locally at each time step. In numerical experiments with a wide range of evolution equations, the proposed scheme with randomized sparse updates is up to two orders of magnitude more accurate at a fixed computational budget and up to two orders of magnitude faster at a fixed accuracy than schemes with dense updates.