UNOP: Physics-Constrained Unsupervised Neural Operator for Long-Horizon PDE Learning on Generalized Geometries
TL;DR
This work presents a physics-constrained un-supervised neural operator for long-horizon PDE learning on generalized geometries (UNOP), which replaces differential constraints with integral consistency for stable, label-free learning.
Abstract
Unsupervised learning of neural operators is constrained by numerical instability, causing predictions to diverge in long-horizon rollouts. To address this, we present a physics-constrained unsupervised neural operator for long-horizon PDE learning on generalized geometries (UNOP). This framework replaces differential constraints with integral consistency for stable, label-free learning. Unlike prior works, UNOP is built upon Latent Integral Physics Embedding (LIPE), which enforces physical consistency through integral constraints. To extend integral formulations to generalized geometries, the Geometry-Agnostic Latent Adapter (GALA) projects them onto a unified latent grid of PDE inputs, providing a regularized domain for spatial integral evaluation. Based on this shared embedding, the Gated Spectral Evolution Operator (GSEO) performs stable temporal integration while retaining spatial regions with sharp gradients and fine-scale structures, with the evolution constrained by the LIPE objective. Experiments on 1D, 2D, and 3D benchmarks show UNOP outperforms state of the art methods, reducing error accumulation by up to 60% in 20-step rollouts. Code is available at https://github.com/chengxinrui/UNOP.