(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable solutions, their accuracy is often tied to the use of computationally intensive fine meshes. Moreover, they do not naturally account for measurements or prior solutions, and any change in the problem parameters requires results to be fully recomputed. Neural network-based approaches, such as physics-informed neural networks and neural operators, offer a mesh-free alternative by directly fitting those models to the PDE solution. They can also integrate prior knowledge and tackle entire families of PDEs by simply aggregating additional training losses. Nevertheless, they are highly sensitive to hyperparameters such as collocation points and the weights associated with each loss. This paper addresses these challenges by developing a science-constrained learning (SCL) framework. It demonstrates that finding a (weak) solution of a PDE is equivalent to solving a constrained learning problem with worst-case losses. This explains the limitations of previous methods that minimize the expected value of aggregated losses. SCL also organically integrates structural constraints (e.g., invariances) and (partial) measurements or known solutions. The resulting constrained learning problems can be tackled using a practical algorithm that yields accurate solutions across a variety of PDEs, neural network architectures, and prior knowledge levels without extensive hyperparameter tuning and sometimes even at a lower computational cost.

翻译：（偏）微分方程是描述自然现象的基本工具，其求解在科学与工程领域至关重要。尽管传统方法（如有限元法）能提供可靠解，但其精度通常依赖于计算密集的精细网格。此外，这些方法无法自然地纳入测量数据或先验解，且问题参数的任何变化都需要完全重新计算结果。基于神经网络的方法（如物理信息神经网络与神经算子）通过直接拟合微分方程解，提供了一种无网格替代方案。它们还能通过简单聚合附加训练损失来整合先验知识并处理整个微分方程族。然而，这类方法对超参数（如配置点及各损失权重）极为敏感。本文通过构建科学约束学习框架应对这些挑战。研究证明，寻找微分方程的（弱）解等价于求解具有最坏情况损失的约束学习问题，这解释了以往最小化聚合损失期望值方法的局限性。该框架还能有机整合结构约束（如不变性）及（部分）测量数据或已知解。由此产生的约束学习问题可通过实用算法求解，该算法能在无需大量超参数调优的情况下，针对各类微分方程、神经网络架构及先验知识水平获得精确解，有时甚至能以更低计算成本实现。