提出统一变分框架，让神经微分方程求解的自适应采样更科学可靠。
A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning
- 用变分法统一残差自适应策略，不同变换对应不同误差目标。
- 自适应采样降低损失估计方差，提升梯度信噪比，减少离散化误差。
- 适用于多种优化器和模型结构，为训练策略提供理论支撑。
残差自适应策略在科学机器学习中广泛应用，但多为启发式方法。本文提出一个统一的变分框架，通过残差的凸变换形式化这些方法。不同的变换对应不同的目标泛函：指数权重旨在最小化一致误差，线性权重则对应二次误差最小化。在此视角下，自适应加权等价于选择最优采样分布以优化主目标，从而将离散化选择与误差度量直接关联。该原则性方法带来三方面优势：(1) 可系统设计跨范数的自适应方案；(2) 通过降低损失估计方差减少离散化误差；(3) 改善梯度信噪比，增强学习动态。将框架扩展至算子学习后，在多种优化器与架构上均实现显著性能提升。结果为残差自适应提供了理论依据，并建立了可解释的离散化与训练策略基础。
原文摘要 · Abstract (English)
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.
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