HyperLoRA for PDEs

TL;DR

This paper introduces HyperLoRA, a physics-informed hypernetwork approach that efficiently predicts neural network weights for solving parameterized PDEs, achieving significant parameter reduction and improved generalization.

Contribution

The paper proposes a novel HyperLoRA method combining hypernetworks with low-rank adaptation for PDE solutions, enhanced by physics-informed training to improve accuracy and efficiency.

Findings

Achieves 8x reduction in prediction parameters.

Maintains accuracy comparable to baseline methods.

Enables fast solutions for complex parameterized PDEs.

Abstract

Physics-informed neural networks (PINNs) have been widely used to develop neural surrogates for solutions of Partial Differential Equations. A drawback of PINNs is that they have to be retrained with every change in initial-boundary conditions and PDE coefficients. The Hypernetwork, a model-based meta learning technique, takes in a parameterized task embedding as input and predicts the weights of PINN as output. Predicting weights of a neural network however, is a high-dimensional regression problem, and hypernetworks perform sub-optimally while predicting parameters for large base networks. To circumvent this issue, we use a low ranked adaptation (LoRA) formulation to decompose every layer of the base network into low-ranked tensors and use hypernetworks to predict the low-ranked tensors. Despite the reduced dimensionality of the resulting weight-regression problem, LoRA-based Hypernetworks violate the underlying physics of the given task. We demonstrate that the generalization capabilities of LoRA-based hypernetworks drastically improve when trained with an additional physics-informed loss component (HyperPINN) to satisfy the governing differential equations. We observe that LoRA-based HyperPINN training allows us to learn fast solutions for parameterized PDEs like Burger's equation and Navier Stokes: Kovasznay flow, while having an 8x reduction in prediction parameters on average without compromising on accuracy when compared to all other baselines.