Physics-Informed Deep B-Spline Networks

TL;DR

This work introduces Physics-Informed Deep B-Spline Networks (PI-BSNet), a framework that learns families of parametric PDEs with varying ICBCs by predicting B-spline control points through a coefficient network. By using a fixed B-spline basis, PI-BSNet enforces initial and boundary conditions by construction and enables analytic derivatives for PDE residual losses, achieving efficient physics-informed learning across parameterized domains. The authors establish universal approximation guarantees for PI-BSNet and derive generalization error bounds for elliptic and parabolic PDEs, addressing theoretical gaps for neural representations based on basis functions. Empirically, PI-BSNet demonstrates improved efficiency-accuracy trade-offs and robustness to nonhomogeneous ICBCs and non-rectangular domains, outperforming several PINN and neural-operator baselines on recovery-probability, advection with irregular ICBCs, and trapezoidal-domain diffusion tasks.

Abstract

Physics-informed machine learning offers a promising framework for solving complex partial differential equations (PDEs) by integrating observational data with governing physical laws. However, learning PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) with theoretical guarantees remains an open challenge. In this paper, we propose physics-informed deep B-spline networks, a novel technique that approximates a family of PDEs with different parameters and ICBCs by learning B-spline control points through neural networks. The proposed B-spline representation reduces the learning task from predicting solution values over the entire domain to learning a compact set of control points, enforces strict compliance to initial and Dirichlet boundary conditions by construction, and enables analytical computation of derivatives for incorporating PDE residual losses. While existing approximation and generalization theories are not applicable in this setting - where solutions of parametrized PDE families are represented via B-spline bases - we fill this gap by showing that B-spline networks are universal approximators for such families under mild conditions. We also derive generalization error bounds for physics-informed learning in both elliptic and parabolic PDE settings, establishing new theoretical guarantees. Finally, we demonstrate in experiments that the proposed technique has improved efficiency-accuracy tradeoffs compared to existing techniques in a dynamical system problem with discontinuous ICBCs and can handle nonhomogeneous ICBCs and non-rectangular domains.

Figures (14)

• Figure 1: Diagram of PI-BSNet. The coefficient network takes system and ICBC parameters as input and outputs the control points tensor, which is then multiplied by the B-spline basis to produce the final output. Physics and data losses are used to train the coefficient network using closed-form gradient formulas, while the compliance of ICBC conditions is strictly enforced through the B-spline basis. Solid lines depict the forward pass, and dashed lines depict the backward pass of the network.
• Figure 2: Recovery probability visualizations. Predictions in first row and errors in second row.
• Figure 3: Losses vs. epochs with mean and standard deviation over 10 independent runs.
• Figure 4: Training time vs. prediction error trade-offs.
• Figure 5: Test results on advection equations with unseen parameter.

Theorems & Definitions (16)

• Theorem 4.3
• Theorem 4.6
• Theorem A.1
• proof
• Lemma A.2
• proof
• Lemma A.3
• proof
• Theorem A.4
• proof