A data-driven, model-free approach to modeling the temporal evolution of physical systems mitigates the need for explicit knowledge of the governing equations. Even when physical priors such as partial differential equations are available, such systems often reside in high-dimensional state spaces and exhibit nonlinear dynamics, making traditional numerical solvers computationally expensive and ill-suited for real-time analysis and control. Consider the problem of learning a parametric flow of a dynamical system: with an initial field and a set of physical parameters, we aim to predict the system's evolution over time in a way that supports long-horizon rollouts, generalization to unseen parameters, and spectral analysis. We propose a physics-coded neural field parameterization of the Koopman operator's spectral decomposition. Unlike a physics-constrained neural field, which fits a single solution surface, and neural operators, which directly approximate the solution operator at fixed time horizons, our model learns a factorized flow operator that decouples spatial modes and temporal evolution. This structure exposes underlying eigenvalues, modes, and stability of the underlying physical process to enable stable long-term rollouts, interpolation across parameter spaces, and spectral analysis. We demonstrate the efficacy of our method on a range of dynamics problems, showcasing its ability to accurately predict complex spatiotemporal phenomena while providing insights into the system's dynamic behavior.

翻译：一种数据驱动的无模型方法用于模拟物理系统的时间演化，减少了对显式控制方程知识的依赖。即使存在偏微分方程等物理先验，此类系统通常处于高维状态空间并呈现非线性动力学特性，使得传统数值求解器计算成本高昂且难以适用于实时分析与控制。考虑学习动力系统参数化流的问题：给定初始场和一组物理参数，我们的目标是预测系统随时间的演化，同时支持长时程推演、对未见参数的泛化以及谱分析。我们提出了一种基于物理编码神经场的Koopman算子谱分解参数化方法。与拟合单一解曲面的物理约束神经场以及直接近似固定时间范围解算子的神经算子不同，我们的模型学习了一个因子化流算子，该算子解耦了空间模态与时间演化。这种结构显式揭示了底层物理过程的本征值、模态和稳定性，从而实现了稳定的长期推演、参数空间插值以及谱分析。我们在多种动力学问题上验证了该方法的有效性，展示了其准确预测复杂时空现象的能力，同时为系统动态行为提供了深入洞察。