PI-Splines Offer Stable Alternative for Physics-Informed Learning

Physics-Informed Neural Networks (PINNs) have gained traction for solving differential equations by embedding physical laws into the training process. This work proposes an alternative architecture called Physics-Informed Splines (PI-Splines). Instead of using a neural network, PI-Splines directly represent the unknown field through a tensor-product B-spline expansion, where the control coefficients are trainable parameters. This formulation maintains the residual-based training paradigm characteristic of PINNs. The PI-Splines approach offers several distinct advantages: it provides compact support, allowing for localized adjustments; it enables explicit control over smoothness; it yields analytical derivatives, which are crucial for physics-informed problems; and its trainable parameters have a direct geometric interpretation. Boundary conditions can also be strongly imposed by fixing specific boundary control coefficients when compatible with the spline representation. Evaluated against standard physics-informed frameworks on various benchmark problems, PI-Splines demonstrate competitive and stable performance, particularly beneficial in scenarios where structured representations, locality, and parameter efficiency are desired.