Trainable Spline Representations for Physics-Informed Learning

2026-07-17

AI Summary

This work addresses key limitations of conventional physics-informed neural networks (PINNs) in solving differential equations—namely parameter redundancy, weak locality, and insufficient control over solution smoothness. The authors propose Physics-Informed Splines (PI-Splines), which directly parameterize the unknown field using tensor-product B-splines with trainable control points. Retaining the residual-driven training paradigm of PINNs, PI-Splines inherently offer compact support, explicit smoothness control, and analytically computable derivatives. By strongly enforcing boundary conditions, the method endows spline parameters with clear geometric meaning while leveraging structured representation and efficient optimization strategies. Experimental results demonstrate that PI-Splines achieve stable and efficient performance across multiple benchmark problems, significantly outperforming traditional neural network architectures in terms of parameter efficiency, locality, and representational capacity.

Abstract

This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.

Problem

• physics-informed learning
• differential equations
• structured representation
• boundary conditions
• approximation architecture

Innovation

• Physics-Informed Splines
• B-spline representation
• trainable control coefficients
• strong boundary enforcement
• parameter efficiency