Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

TL;DR

The paper tackles instability in neural operators for PDEs during long-horizon rollout by enforcing geometry-aware latent dynamics. It introduces the Manifold-Constrained Layer (MCL), which uses Lie group constraints via a low-rank Lie algebra to produce norm-preserving, near-isometric updates that plug into existing neural operators. Across multiple PDEs (e.g., Burgers, Darcy, Navier-Stokes), MCL yields 30-50% reductions in rollout error with only about a 2.26% parameter overhead, improving long-term fidelity and physical consistency. This geometry-aware approach offers a scalable, plug-in enhancement with potential impact on climate modeling, fluid dynamics, and engineering simulations.

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.

Figures (10)

• Figure 1: Conceptual comparison between common residual updates in existing neural operators vs. our geometric constrained updates. Our method, termed MCL, addresses the challenge of physical law violation in neural operator update by enforcing geometric inductive bias through Lie algebra parameterization.
• Figure 2: Framework of our method, termed MCL, when integrated in a neural operator. MCL replaces the standard unconstrained Euclidean space-based update with geometric-constrained updates based on Lie algebra group action. MCL serves as an efficient plug-and-play module for existing neural operators.
• Figure 3: Prediction error plot of PDEs on FNO (left) and FNO+MCL (right) against true solution.
• Figure 4: Evolution of energy (left) and entropy (right) on 1-D Burgers equation with parameter values 0.001 (top), 0.01 (middle), and 0.1 (bottom).
• Figure A1: Plots on true solution, prediction on NO vs. NO+MCL, and the corresponding error plots on 1-D linear advection equation. Models used for prediction are, from top to bottom, FNO, U-NO, and GINO.

Theorems & Definitions (1)

• Proposition A.1: Invertibility of Near-Isometric Updates