Differentiability in Unrolled Training of Neural Physics Simulators on Transient Dynamics

TL;DR

This paper investigates how different unrolling training methods affect neural physics simulators' accuracy, revealing that non-differentiable unrolling with numerical solvers can outperform fully differentiable training in certain scenarios.

Contribution

The study provides a comprehensive empirical analysis of unrolled training variants, highlighting the benefits of non-differentiable unrolling combined with numerical solvers for physics simulation accuracy.

Findings

Non-differentiable unrolling with solvers can outperform fully differentiable training.

Differentiable training yields the highest accuracy among correction and prediction setups.

Behavior is consistent across physical systems, architectures, and numerical schemes.

Abstract

Unrolling training trajectories over time strongly influences the inference accuracy of neural network-augmented physics simulators. We analyze this in three variants of training neural time-steppers. In addition to one-step setups and fully differentiable unrolling, we include a third, less widely used variant: unrolling without temporal gradients. Comparing networks trained with these three modalities disentangles the two dominant effects of unrolling, training distribution shift and long-term gradients. We present detailed study across physical systems, network sizes and architectures, training setups, and test scenarios. It also encompasses two simulation modes: In prediction setups, we rely solely on neural networks to compute a trajectory. In contrast, correction setups include a numerical solver that is supported by a neural network. Spanning these variations, our study provides the empirical basis for our main findings: Non-differentiable but unrolled training with a numerical solver in a correction setup can yield substantial improvements over a fully differentiable prediction setup not utilizing this solver. The accuracy of models trained in a fully differentiable setup differs compared to their non-differentiable counterparts. Differentiable ones perform best in a comparison among correction networks as well as among prediction setups. For both, the accuracy of non-differentiable unrolling comes close. Furthermore, we show that these behaviors are invariant to the physical system, the network architecture and size, and the numerical scheme. These results motivate integrating non-differentiable numerical simulators into training setups even if full differentiability is unavailable. We show the convergence rate of common architectures to be low compared to numerical algorithms. This motivates correction setups combining neural and numerical parts which utilize benefits of both.