Physics-informed neural networks (PINNs) have achieved notable success in modeling dynamical systems governed by partial differential equations (PDEs). To avoid computationally expensive retraining under new physical conditions, parameterized PINNs (P$^2$INNs) commonly adapt pre-trained operators using singular value decomposition (SVD) for out-of-distribution (OOD) regimes. However, SVD-based fine-tuning often suffers from rigid subspace locking and truncation of important high-frequency spectral modes, limiting its ability to capture complex physical transitions. While parameter-efficient fine-tuning (PEFT) methods appear to be promising alternatives, applying conventional adapters such as LoRA to P$^2$INNs introduces a severe Pareto trade-off, as additive updates increase parameter overhead and disrupt the structured physical manifolds inherent in operator representations. To address these limitations, we propose Manifold-Orthogonal Dual-spectrum Extrapolation (MODE), a lightweight micro-architecture designed for physics operator adaptation. MODE decomposes physical evolution into complementary mechanisms including principal-spectrum dense mixing that enables cross-modal energy transfer within frozen orthogonal bases, residual-spectrum awakening that activates high-frequency spectral components through a single trainable scalar, and affine Galilean unlocking that explicitly isolates spatial translation dynamics. Experiments on challenging PDE benchmarks including the 1D Convection--Diffusion--Reaction equation and the 2D Helmholtz equation demonstrate that MODE achieves strong out-of-distribution generalization while preserving the minimal parameter complexity of native SVD and outperforming existing PEFT-based baselines.

翻译：物理信息神经网络（PINNs）在模拟由偏微分方程（PDE）控制的动力系统方面取得了显著成功。为避免在新物理条件下进行昂贵的重训练，参数化物理信息神经网络（P$^2$INNs）通常采用奇异值分解（SVD）对预训练算子进行分布外（OOD）适应。然而，基于SVD的微调常受限于刚性子空间锁定及重要高频谱模式截断，导致其难以捕捉复杂物理变迁。尽管参数高效微调（PEFT）方法看似具有潜力，将LoRA等传统适配器应用于P$^2$INNs会引入严重的帕累托权衡——加性更新不仅增加参数开销，更会破坏算子表示中固有的结构化物理流形。针对上述局限，我们提出流形正交双谱外推（MODE）方法，这是一种专为物理算子适配设计的轻量级微架构。MODE将物理演化分解为互补机制：主谱密集混合实现冻结正交基内的跨模态能量传递；残谱唤醒通过单个可训练标量激活高频谱分量；仿射伽利略解锁则显式分离空间平移动力学。在包含一维对流-扩散-反应方程与二维亥姆霍兹方程等挑战性PDE基准上的实验表明，MODE在保持原生SVD最小参数复杂度的同时，实现了强分布外泛化性能，并超越现有基于PEFT的基线方法。