通过梯度对齐检测神经网络是否内化物理对称性，揭示模型泛化机制。
Loss Landscape Geometry of Partial Differential Equation Emulators: Or, Symmetry Learning via Gradient Alignment
- 用群轨道上梯度重叠衡量参数更新传播，评估对称性学习
- 发现梯度一致性促进模型对称泛化，指引训练进入对称基域
- 适用于流体模拟等偏微分方程代理模型的对称性验证
我们研究神经网络在模拟偏微分方程解算子时如何内化物理对称性，提出一种基于影响的诊断方法：通过度量沿群轨道计算的损失梯度在度量加权下的重叠程度，来评估参数更新在对称相关状态间的传播。该指标探查了学习损失景观的局部几何结构，超越了前向传递的等变性测试，直接判断学习动态是否耦合物理等价配置。应用于自回归流体流动代理模型，结果表明轨道级梯度一致性是模型实现对称变换泛化的机制，并指示训练是否选择与对称性兼容的极小值区域。此方法为评估代理模型是否内化已知解算子的对称性质提供了新手段。
原文摘要 · Abstract (English)
We study how neural emulators of partial differential equation solution operators internalize physical symmetries by introducing an influence-based diagnostic that measures the propagation of parameter updates between symmetry-related states, defined as the metric-weighted overlap of loss gradients evaluated along group orbits. This quantity probes the local geometry of the learned loss landscape and goes beyond forward-pass equivariance tests by directly assessing whether learning dynamics couple physically equivalent configurations. Applying our diagnostic to autoregressive fluid flow emulators, we show that orbit-wise gradient coherence provides the mechanism for learning to generalize over symmetry transformations and indicates when training selects a symmetry compatible basin. The result is a novel technique for evaluating if surrogate models have internalized symmetry properties of the known solution operator.
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