Stochastic Taylor Derivative Estimator: Efficient amortization for arbitrary differential operators

TL;DR

This paper introduces an efficient method for computing high-order derivatives of multivariate functions, significantly speeding up neural network training involving complex differential operators, especially in large-scale physics-informed problems.

Contribution

It presents a novel approach to perform arbitrary contraction of derivative tensors using high-order auto-differentiation, enabling efficient randomization for multivariate functions.

Findings

>1000× speed-up in PINNs training

>30× memory reduction compared to first-order AD

Solved 1-million-dimensional PDEs in 8 minutes on a single GPU

Abstract

Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to $\mathcal{O}(d^{k})$ scaling of the derivative tensor size and the $\mathcal{O}(2^{k-1}L)$ scaling in the computation graph, where $d$ is the dimension of the domain, $L$ is the number of ops in the forward computation graph, and $k$ is the derivative order. In previous works, the polynomial scaling in $d$ was addressed by amortizing the computation over the optimization process via randomization. Separately, the exponential scaling in $k$ for univariate functions ($d=1$) was addressed with high-order auto-differentiation (AD). In this work, we show how to efficiently perform arbitrary contraction of the derivative tensor of arbitrary order for multivariate functions, by properly constructing the input tangents to univariate high-order AD, which can be used to efficiently randomize any differential operator. When applied to Physics-Informed Neural Networks (PINNs), our method provides >1000$\times$ speed-up and >30$\times$ memory reduction over randomization with first-order AD, and we can now solve \emph{1-million-dimensional PDEs in 8 minutes on a single NVIDIA A100 GPU}. This work opens the possibility of using high-order differential operators in large-scale problems.