Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

• URL: http://arxiv.org/abs/2602.16209v1
• Date: Wed, 18 Feb 2026 06:17:47 GMT
• Title: Geometric Neural Operators via Lie Group-Constrained Latent Dynamics
• Authors: Jiaquan Zhang, Fachrina Dewi Puspitasari, Songbo Zhang, Yibei Liu, Kuien Liu, Caiyan Qin, Fan Mo, Peng Wang, Yang Yang, Chaoning Zhang,
• Abstract summary: We show that our method effectively lowers the relative prediction error by 30-50% at the cost of 2.26% of parameter increase.<n>The results show that our approach provides a scalable solution for improving long-term prediction fidelity.
• Score: 14.152015935335358
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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