Neural Manifold Operators for Learning the Evolution of Physical Dynamics

Modeling the evolution of physical dynamics is a foundational problem in science and engineering, and it is regarded as the modeling of an operator mapping between infinite-dimensional functional spaces. Operator learning methods, learning the underlying infinite-dimensional operator in a high-dimensional latent space, have shown significant potential in modeling physical dynamics. However, there remains insufficient research on how to approximate an infinite-dimensional operator using a finite-dimensional parameter space. Inappropriate dimensionality representation of the underlying operator leads to convergence difficulties, decreasing generalization capability, and violating the physical consistency. To address the problem, we present Neural Manifold Operator (NMO) to learn the invariant subspace with the intrinsic dimension to parameterize infinite-dimensional underlying operators. NMO achieves state-of-the-art performance in statistical and physical metrics and gains 23.35% average improvement on three real-world scenarios and four equation-governed scenarios across a wide range of multi-disciplinary fields. Our paradigm has demonstrated universal effectiveness across various model structure implementations, including Multi-Layer Perceptron, Convolutional Neural Networks, and Transformers. Experimentally, we prove that the intrinsic dimension calculated by our paradigm is the optimal dimensional representation of the underlying operators. We release our code at https://github.com/AI4EarthLab/Neural-Manifold-Operators.