Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks

TL;DR

This paper introduces a neural Galerkin scheme that updates randomized sparse subsets of network parameters at each time step, significantly improving efficiency and accuracy in solving evolution equations.

Contribution

The work proposes a novel randomized sparse update method for neural Galerkin schemes, reducing computational costs and error accumulation in time-dependent PDE solutions.

Findings

Up to 100x more accurate at fixed computational cost

Up to 100x faster at fixed accuracy

Effective across various evolution equations

Abstract

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.