Spatio-Temporal Neural Operator for Complex Riemannian Geometries
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Solution Overview
Problem
Existing neural operator networks struggle with accurately learning spatio-temporal functions on complex geometries due to high-dimensional discretization and limitations in handling non-uniform grid data, particularly in the temporal dimension, which affects their computational efficiency and accuracy.
Innovation Solution
A spatio-temporal neural operator on Riemannian manifolds is constructed using Laplacian eigenfunctions to encode spatial and temporal dimensions independently, allowing for a nested kernel integration module that learns the mapping between spatio-temporal functions on complex geometries, independent of discretization resolution.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If DeepONet is used to learn spatio-temporal functions, then the model can handle high-dimensional mapping problems, but the number of network parameters still depends on the discretization resolution of the input domain, increasing device complexity
Solution Approach 1:
The patent transforms the spatio-temporal function learning problem from direct high-dimensional spatial mapping into a two-stage process: first mapping spatial coordinates to temporal frequencies, then mapping temporal frequencies to output values. This dimensionality change separates the spatial and temporal dimensions, allowing the model to handle high-dimensional input without proportionally increasing parameters, as the temporal dimension is compressed into a frequency spectrum representation.
2Device complexity
If Fourier transform or wavelet transform is used to reduce model complexity, then the mapping complexity is greatly reduced, but uniform grid data defined on a rectangular domain is required, reducing adaptability to complex geometries
Solution Approach 1:
The patent applies local quality by using Laplacian eigenfunctions that are specifically adapted to the geometry of each domain. Instead of using a universal transform like Fourier that requires uniform grids, the method computes geometry-specific eigenfunctions that capture the local structural characteristics of complex domains, allowing the model to maintain low complexity while adapting to arbitrary geometries.
Solution Approach 2:
The patent changes the fundamental parameters of the transform by replacing the standard Fourier basis with Laplacian eigenfunction bases that are tailored to the specific geometry. This parameter change allows the model to work with non-uniform grids and complex geometries while maintaining the computational efficiency of spectral methods.
3Measurement precision
If NORM is used to encode complex geometric information with Laplacian eigenfunctions, then the spatial dimension mapping is effective, but high-dimensional discretization remains in the temporal dimension, increasing device complexity
Solution Approach 1:
The patent applies dimensionality change by transforming the temporal dimension from a high-dimensional discrete sequence into a low-dimensional frequency spectrum. Instead of processing temporal data point-by-point in high dimensions, the method projects temporal variations onto a small number of dominant frequency modes, dramatically reducing the temporal discretization complexity while preserving essential temporal dynamics.
Data Source
AI summary
A construction method of spatio-temporal neural operator on Riemannian manifolds for complex geometries is disclosed, and relates to the field of machine learning technology, the method comprises the following steps: S1, according to a geometric space of an input spatio-temporal function and an output spatio-temporal function, solving a set of Laplacian eigenfunctions as basis functions, and then constructing a spatial dimension encoding module and a spatial dimension decoding module ; S2, solving a set of Fourier basis functions, and then constructing a temporal dimension encoding module and a temporal dimension decoding module −1; and S3, constructing a Laplace-Fourier nested kernel integration module, and constructing a spatio-temporal neural operator on Riemannian manifolds for complex geometries by serially connecting a plurality of kernel integration modules. The present disclosure adopts the construction method of spatio-temporal neural operator on Riemannian manifolds for complex geometries based on the above steps, and the mapping between two spatio-temporal functions defined on complex geometries is represented by constructing a parameterized model.