发表机构
Courant Institute School of Mathematics, Computing and Data Science, New York University（纽约大学柯朗数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出ICNNE，一种输入凸神经网络架构，通过学习耗散势而非本构定律，从速度数据中保证PDE良置性，并在多个流体与海冰问题中实现准确且可泛化的物理学习。
AI 中文摘要
从速度数据（即间接观测）学习复杂流体的本构定律是一个受PDE约束的反问题，其中表达性强的神经参数化可能破坏正向物理模型的良置性。我们通过学习耗散势而非本构定律本身来解决这一矛盾：耗散势是一个标量函数，其凸性、框架无差异性和耗散性会传播到基础连续介质力学中，并保证良置且可泛化的正向PDE所需的结构性质。我们引入ICNNE，一种输入凸神经网络架构，通过对称化精确强制第二应变率不变量的凸性、框架无差异性和偶性，缓解了标准公式在小应变率下出现的数值不稳定性。通过对势的原点零梯度进行弱惩罚，ICNNE还满足耗散性。损失目标及其梯度通过有限元和神经网络方法的组合计算，耦合了Firedrake和PyTorch库。我们在四个跨越可压缩和不可压缩区域的问题上进行了评估：可压缩Navier-Stokes、Herschel-Bulkley屈服应力流、Hibler的粘塑性海冰模型，以及没有已知本构定律的离散元方法数据。我们表明，学习到的势（i）在已知物理的地方恢复真实物理，（ii）准确迁移到训练中未见过的几何形状，（iii）在非结构化方法发散的区域取得成功。这些结果表明，将数学结构嵌入参数化而非损失中，是从间接数据学习物理的稳健路径。
英文摘要
Learning constitutive laws of complex fluids from velocity data (i.e. indirect observations) is a PDE-constrained inverse problem in which expressive neural parameterizations risk breaking the well-posedness of the forward physics model. We address this tension by learning, rather than the constitutive law itself, the dissipation potential: a scalar function whose convexity, frame-indifference, and dissipativity propagate into the underlying continuum mechanics and guarantee the structural properties needed for a well-posed and generalizable forward PDE. We introduce ICNNE, an input-convex neural architecture that exactly enforces convexity, frame-indifference, and evenness in the second strain-rate invariant via symmetrization, alleviating the numerical instabilities that occur at small strain rates in standard formulations. By weakly enforcing zero gradients in the origin of the potential via penalization, ICNNE also satisfies the dissipativity property. The loss objective and its gradient are computed using a combination of finite element and neural network methods, coupling the Firedrake and PyTorch libraries. We evaluate on four problems spanning compressible and incompressible regimes: compressible Navier-Stokes, Herschel-Bulkley yield-stress flow, Hibler's viscous-plastic sea-ice model, and discrete-element-method data with no known constitutive law. We show that the learned potentials (i) recover ground-truth physics where it is known, (ii) transfer accurately to geometries unseen during training, and (iii) succeed in regimes where unstructured methods diverge. These results suggest that embedding mathematical structure into the parameterization, rather than into the loss, is a robust path to learning physics from indirect data.
