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.

翻译：物理信息机器学习为求解复杂偏微分方程提供了一个前景广阔的框架，它通过将观测数据与支配物理定律相结合来实现这一目标。然而，在理论保证下学习具有变化参数以及时变初始条件和边界条件的偏微分方程，仍然是一个开放的挑战。本文提出了一种物理信息深度B样条网络，这是一种新颖的技术，它通过神经网络学习B样条控制点，来近似具有不同参数和初始/边界条件的偏微分方程族。所提出的B样条表示将学习任务从预测整个域上的解值简化为学习一组紧凑的控制点，通过其构造方式强制严格满足初始条件和狄利克雷边界条件，并支持解析计算导数以纳入偏微分方程残差损失。虽然现有的近似理论和泛化理论不适用于这种通过B样条基表示参数化偏微分方程族解的场景，但我们填补了这一空白，证明了在温和条件下，B样条网络是此类方程族的通用逼近器。我们还推导了在椭圆型和抛物型偏微分方程设定下物理信息学习的泛化误差界，从而建立了新的理论保证。最后，我们在实验中证明，在一个具有不连续初始/边界条件的动力系统问题中，与现有技术相比，所提出的技术在效率-精度权衡方面有所改进，并且能够处理非齐次初始/边界条件以及非矩形域。