Title: A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning Show Chinese title

Title: 基于残差的神经微分方程求解器和算子学习自适应的变分框架

Abstract: Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the residual. Different transformations correspond to distinct objective functionals: exponential weights target the minimization of uniform error, while linear weights recover the minimization of quadratic error. Within this perspective, adaptive weighting is equivalent to selecting sampling distributions that optimize the primal objective, thereby linking discretization choices directly to error metrics. This principled approach yields three benefits: (1) it enables systematic design of adaptive schemes across norms, (2) reduces discretization error through variance reduction of the loss estimator, and (3) enhances learning dynamics by improving the gradient signal-to-noise ratio. Extending the framework to operator learning, we demonstrate substantial performance gains across optimizers and architectures. Our results provide a theoretical justification of residual-based adaptivity and establish a foundation for principled discretization and training strategies. Show Chinese abstract

Abstract: 基于残差的自适应策略在科学机器学习中被广泛使用，但仍然主要是一种启发式方法。 我们引入了一个统一的变分框架，通过整合残差的凸变换来形式化这些方法。 不同的变换对应于不同的目标泛函：指数权重旨在最小化均匀误差，而线性权重则恢复二次误差的最小化。 从这个角度来看，自适应加权等价于选择优化原始目标的采样分布，从而将离散化选择直接与误差度量联系起来。 这种有原则的方法带来了三个好处：(1) 它使得在不同范数下系统地设计自适应方案成为可能，(2) 通过减少损失估计器的方差来降低离散化误差，(3) 通过提高梯度信噪比来增强学习动力学。 将该框架扩展到算子学习，我们在不同的优化器和架构上展示了显著的性能提升。 我们的结果为基于残差的自适应性提供了理论依据，并建立了有原则的离散化和训练策略的基础。