Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

TL;DR

This paper introduces MCL, a Lie group-based manifold constraint method that enhances neural operators for PDE solutions by enforcing geometric laws, significantly improving stability and accuracy in long-term predictions.

Contribution

The paper proposes a novel Lie group-constrained latent dynamics approach that acts as a plug-and-play module to improve neural operators' stability and accuracy for PDEs.

Findings

Reduces prediction error by 30-50% across tested PDEs.

Increases model parameters by only 2.26%.

Enhances long-term prediction fidelity with geometric constraints.

Abstract

Neural operators offer an effective framework for learning solutions of partial differential equations for many physical systems in a resolution-invariant and data-driven manner. Existing neural operators, however, often suffer from instability in multi-layer iteration and long-horizon rollout, which stems from the unconstrained Euclidean latent space updates that violate the geometric and conservation laws. To address this challenge, we propose to constrain manifolds with low-rank Lie algebra parameterization that performs group action updates on the latent representation. Our method, termed Manifold Constraining based on Lie group (MCL), acts as an efficient \emph{plug-and-play} module that enforces geometric inductive bias to existing neural operators. Extensive experiments on various partial differential equations, such as 1-D Burgers and 2-D Navier-Stokes, over a wide range of parameters and steps demonstrate that our method effectively lowers the relative prediction error by 30-50\% at the cost of 2.26\% of parameter increase. The results show that our approach provides a scalable solution for improving long-term prediction fidelity by addressing the principled geometric constraints absent in the neural operator updates.