用振荡状态空间模型提升物理神经网络求解偏微分方程的精度与效率
Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers
- 引入基于线性振荡器的时序演化机制，显式建模解的模态结构
- 在100维空间问题上实现更高精度且内存消耗显著降低
- 适合需高效求解高维或长时间演化物理问题的研究者
求解时变偏微分方程（PDE）是计算科学与工程中的关键问题。物理信息神经网络（PINNs）通过控制方程学习解，但准确捕捉时间演化仍具挑战。现有基于序列模型的方法虽能建模时序依赖，却未显式编码解的结构化动态，且内存随序列长度和分辨率增长过快，限制其在大规模或高维场景的应用。本文提出一种新方法，将振荡状态空间动力学融入PINN，通过基于线性振荡器的时序演化与空间上的物理感知谱基结合，实现闭式空间微分并一致施加边界条件。在前向、反向及高维PDE问题上评估，包括最高达100维的空间场景，结果表明该方法在精度和内存效率方面均优于近期基于序列模型的PINN方法。研究表明，在神经PDE求解器中引入结构化动力学先验可提升性能，并为设计更符合物理规律且计算高效的架构提供新思路。
原文摘要 · Abstract (English)
Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.
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