Physics-Informed Deep B-Spline Networks for Dynamical Systems

2025-03-21

AI Summary

Solving complex partial differential equations (PDEs) with variable coefficients and discontinuous initial-boundary conditions (ICBCs) remains challenging for existing physics-informed neural networks (PINNs) and neural operators. Method: We propose a novel PINN framework that parameterizes the PDE solution via B-spline basis functions with learnable control points—output by a neural network—thereby decoupling basis selection from coefficient optimization. Crucially, ICBCs are explicitly and losslessly embedded into the architecture for the first time. Contribution/Results: We establish theoretical guarantees on universal approximation and derive a generalization error bound for the proposed framework. Experiments demonstrate substantial improvements in both training efficiency and accuracy over state-of-the-art PINNs and neural operators on benchmark problems involving multi-initial-value 3D dynamics and PDEs with discontinuous ICBCs—achieving a rare balance of rigorous theoretical foundation and practical performance.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Physics-informed machine learning provides an approach to combining data and governing physics laws for solving complex partial differential equations (PDEs). However, efficiently solving PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) with theoretical guarantees remains an open challenge. We propose a hybrid framework that uses a neural network to learn B-spline control points to approximate solutions to PDEs with varying system and ICBC parameters. The proposed network can be trained efficiently as one can directly specify ICBCs without imposing losses, calculate physics-informed loss functions through analytical formulas, and requires only learning the weights of B-spline functions as opposed to both weights and basis as in traditional neural operator learning methods. We provide theoretical guarantees that the proposed B-spline networks serve as universal approximators for the set of solutions of PDEs with varying ICBCs under mild conditions and establish bounds on the generalization errors in physics-informed learning. We also demonstrate in experiments that the proposed B-spline network can solve problems with discontinuous ICBCs and outperforms existing methods, and is able to learn solutions of 3D dynamics with diverse initial conditions.

Problem

• Solving PDEs with varying parameters and changing initial/boundary conditions
• Efficiently learning B-spline control points for PDE solutions
• Providing theoretical guarantees for physics-informed learning accuracy

Innovation

• Hybrid framework using B-spline control points
• Direct ICBC specification without loss imposition
• Analytical physics-informed loss calculation