Physics-Informed Deep B-Spline Networks
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Abstract (arXiv)
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.
Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 43 of its 98 references that are in the map. First posted on arXiv 21 Mar 2025.
Builds on
43 of 98 references.
References · 43
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2024KAN: Kolmogorov-Arnold Networkscited by 649
- 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
- 2017Solving high-dimensional partial differential equations using deep learningcited by 556
- 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
- 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
- 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
- 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
- 2019Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networkscited by 293
- 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
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- 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 174
- 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
- 2022Physics-Informed Deep Neural Operator Networkscited by 113
- 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 102
- 2021CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Methodcited by 93
- 2019Physics-Informed Neural Networks for Power Systemscited by 80
- 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
- 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 57
- 2021HyperPINN: Learning parameterized differential equations with physics-informed hypernetworkscited by 39
- 2021Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNscited by 38
- 2024Parameterized Physics-informed Neural Networks for Parameterized PDEscited by 38
- 2021Meta-Auto-Decoder for Solving Parametric Partial Differential Equationscited by 36
- 2022Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)cited by 35
- 2022ExSpliNet: An interpretable and expressive spline-based neural networkcited by 33
- 2023Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionalscited by 26
- 2022Meta-PDE: Learning to Solve PDEs Quickly Without a Meshcited by 18
- 2020Accelerating Physics-Informed Neural Network Training with Prior Dictionariescited by 17
- 2024Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solvingcited by 17
- 2022M2N: Mesh Movement Networks for PDE Solverscited by 16
- 2024VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behaviorcited by 13
- 2022A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEscited by 12
- 2024Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methodscited by 10
- 2023Improving physics-informed DeepONets with hard constraintscited by 9
- 2021Safe Control in the Presence of Stochastic Uncertaintiescited by 6
- 2021Differentiable Spline Approximationscited by 4
- 2023A Generalizable Physics-informed Learning Framework for Risk Probability Estimationcited by 4
- 2022Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functionscited by 3
- 2024Building Hybrid B-Spline And Neural Network Operatorscited by 2
- 2024Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systemscited by 2
- 2024Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximantscited by 2
- 2013A note on the continuity of free-boundaries in finite-horizon optimal stopping problems for one dimensional diffusionscited by 1
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