用神经基函数方法精准求解多尺度渗流问题，提升模型稳定性和可学习性。
Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method
- 基于物理约束的神经基函数空间与残差度量耦合，实现确定性优化
- 单次求解即得高精度结果，参数化推断速度提升显著
- 适合需要高稳定性与可解释性的物理建模场景
物理驱动模型与机器学习结合用于加速预测日益普遍，但多数‘物理信息’方法将控制方程作为惩罚项处理，其尺度和意义依赖启发式平衡，模糊了算子结构，导致解逼近误差与方程强制误差混淆，使求解与学习过程难以解释和控制。本文提出神经基函数方法，通过预定义的物理相容神经基函数空间与算子诱导残差度量耦合，实现条件良好的确定性最小化。稳定性与可靠性取决于该度量：残差不仅是优化目标，更是与逼近和强制相关的可计算验证指标，在基函数扩充下仍保持稳定，并生成可在参数实例间学习的低维坐标。以对流多尺度达西动力学为具体范例，展示了该方法的普适价值。本方法在单次求解中产生准确稳健的解，并支持高效的参数化推理与算子学习。
原文摘要 · Abstract (English)
Physics-governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics--informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing-equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection-based formulation that couples a predefined, physics-conforming neural basis space with an operator-induced residual metric to obtain a well-conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。