Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

翻译：物理知情神经网络天然面临任务干扰问题，因其依赖共享参数空间同时满足控制微分方程与边界条件。我们利用Fisher信息矩阵量化物理约束模型的有效自由度（$d_{eff}$），借此分析这一结构性冲突。与经典$d_{eff}$（衡量数据相对于统计先验所约束的参数方向数量）不同，本文提出的$d_{eff}$衡量微分算子未约束的参数方向维度。对于具有有限维核的算子，我们证明$d_{eff}$精确收敛至核维度，且与网络宽度、深度或激活函数无关，将其从拟合诊断指标重新诠释为底层连续算子的结构不变量。对于无限维核算子，$d_{eff}$转而测量网络对该核的有限维表示带宽，而非恢复整数型不变量。值得注意的是，$d_{eff}$还可作为先验结构诊断指标——将适定问题的$d_{eff}$驱至零，可验证物理与边界约束已完全吸收网络的自由方向。基于这一表征，我们引入边界适配的子空间投影策略：无需从头重新训练，而是将参数更新投影至预训练物理算子的零空间，使得新边界条件在不干扰已学习物理规律的条件下得到满足。基于梯度的微调虽可达到或超越该效果，但需更多挂钟时间与调参，而子空间投影仅需数秒至数分钟即可实现近乎等效的质量。我们在线性和非线性算子上的验证表明，该方法能精确适配初始/边界偏移及未预见的约束类型。