Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

2026-02-18

AI Summary

This work addresses the instability of existing neural operators during multi-layer iterations and long-term rollouts, which arises from the lack of geometric and conservation constraints in Euclidean latent spaces. To mitigate this, the authors propose the MCL module, which constrains latent representations to a Lie group manifold via low-rank Lie algebra parameterization. This plug-and-play module injects geometric inductive bias into neural operators, ensuring that updates inherently respect underlying physical structures. Evaluated on canonical PDE tasks—including 1-D Burgers’ equation and 2-D Navier–Stokes equations—the method reduces relative prediction errors by 30–50% while introducing only a 2.26% increase in parameter count, substantially improving both long-term rollout stability and structural consistency.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Learning to Search
• Multiagent Systems: Adversarial Agents

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• neural operators
• geometric constraints
• latent dynamics
• long-horizon prediction
• Lie group

Innovation

• neural operators
• Lie group
• geometric constraints
• latent dynamics
• manifold learning