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[2602.16209] Geometric Neural Operators via Lie Group-Constrained Latent Dynamics
Summary
This paper presents a novel approach to neural operators, addressing instability in multi-layer iterations and long-horizon predictions by constraining latent dynamics using Lie group parameterization.
Why It Matters
The research is significant as it enhances the reliability of neural operators in solving partial differential equations, which are crucial in various physical systems. By incorporating geometric constraints, the proposed method improves prediction accuracy while maintaining efficiency, making it relevant for researchers and practitioners in machine learning and applied mathematics.
Key Takeaways
- Introduces Manifold Constraining based on Lie group (MCL) for neural operators.
- Addresses instability in predictions by enforcing geometric constraints.
- Demonstrates a 30-50% reduction in prediction error across various PDEs.
- Maintains a low parameter increase of only 2.26%.
- Provides a scalable solution for long-term prediction fidelity.
Computer Science > Machine Learning arXiv:2602.16209 (cs) [Submitted on 18 Feb 2026] 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 View a PDF of the paper titled Geometric Neural Operators via Lie Group-Constrained Latent Dynamics, by Jiaquan Zhang and 9 other authors View PDF HTML (experimental) 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...
