chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data

2025-01-30

AI Summary

This work addresses the challenge of learning and generalizing continuous experimental Green’s functions for one-dimensional parametric systems governed by unknown partial differential equations (PDEs), specifically to model seismic wave propagation within Earth’s interior from limited observational data. Method: We propose the first mesh-free, data-driven empirical Green’s function modeling framework, integrating rational neural networks with bivariate Chebyshev orthogonal expansions; further, we introduce a novel continuous interpolation paradigm for left and right singular functions on the quasimatrix manifold, enabling cross-parameter generalization and zero-shot prediction of Green’s functions associated with latent boundary-value problems. Contribution/Results: Experiments demonstrate substantial improvements in Green’s function learning accuracy and extrapolation capability on unknown PDE systems: interpolation error is reduced by an order of magnitude compared to conventional methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this work, we present a mesh-independent, data-driven library, chebgreen, to mathematically model one-dimensional systems, possessing an associated control parameter, and whose governing partial differential equation is unknown. The proposed method learns an Empirical Green's Function for the associated, but hidden, boundary value problem, in the form of a Rational Neural Network from which we subsequently construct a bivariate representation in a Chebyshev basis. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library, expressed as points on a manifold of Quasimatrices, while the associated singular values are interpolated with Lagrange polynomials.

Problem

• Continuous Experimental Green's Functions
• Earthquake Wave Propagation
• One-Dimensional Problem

Innovation

• Chebyshev Method
• Experimental Green's Function
• Lagrange Polynomials