发表机构
SISSA; FAST Computing Srl（国际高等研究院; 快速计算公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出物理信息样条（PI - 样条）架构用于物理信息学习，通过张量积B - 样条展开对未知场参数化，保留相关训练范式并具多种优势。在多基准问题测试中与标准框架对比，结果显示其在特定场景下是有竞争力且稳定的替代方案。
AI 中文摘要
本文介绍了物理信息样条（PI - 样条），一种用于物理信息学习的基于样条的结构化架构。PI - 样条不是用神经网络表示微分方程的解，而是通过具有可训练控制系数的张量积B - 样条展开直接对未知场进行参数化。这种公式保留了基于残差的物理信息神经网络训练范式，同时提供紧凑支撑、显式平滑控制、解析导数以及可训练参数的直接几何解释。当与样条表示兼容时，可通过固定合适的边界控制系数来强施加边界条件。该方法在几个难度递增的基准问题上进行评估，并与标准物理信息框架在匹配的控制方程、配置集、损失项和优化程序下进行比较，以分离近似架构的影响。数值实验表明，PI - 样条为神经物理信息架构提供了有竞争力且稳定的替代方案，特别是在需要结构化表示、局部性和参数效率的设置中。
英文摘要
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.
Comments16 pages, 6 figures, 4 tables
