Parameterized partial differential equations (PDEs) underpin the mathematical modeling of complex systems in diverse domains, including engineering, healthcare, and physics. A central challenge in using PDEs for real-world applications is to accurately infer the parameters, particularly when the parameters exhibit non-linear and spatiotemporal variations. Existing parameter estimation methods, such as sparse identification, physics-informed neural networks (PINNs), and neural operators, struggle in such cases, especially with nonlinear dynamics, multiphysics interactions, or limited observations of the system response. To address these challenges, we introduce Neptune, a general-purpose method capable of inferring parameter fields from sparse measurements of system responses. Neptune employs independent coordinate neural networks to continuously represent each parameter field in physical space or in state variables. Across various physical and biomedical problems, where direct parameter measurements are prohibitively expensive or unattainable, Neptune significantly outperforms existing methods, achieving robust parameter estimation from as few as 45 measurements, reducing parameter estimation errors by two orders of magnitude and dynamic response prediction errors by a factor of ten to baselines such as PINNs and neural operators. More importantly, it exhibits superior physical extrapolation capabilities, enabling reliable predictions in regimes far beyond the training data. By facilitating reliable and data-efficient parameter inference, Neptune promises broad transformative impacts in engineering, healthcare, and beyond.

翻译：参数化偏微分方程是工程、医疗健康、物理学等领域复杂系统数学建模的基础。利用偏微分方程解决实际问题的核心挑战在于精确推断参数，尤其是当参数表现出非线性和时空变化时。现有参数估计方法（如稀疏辨识、物理信息神经网络和神经算子）在此类场景中面临困难，特别是在处理非线性动力学、多物理场相互作用或系统响应观测有限的情况下。为解决这些挑战，我们提出Neptune——一种能从稀疏系统响应测量中推断参数场的通用方法。Neptune采用独立坐标神经网络在物理空间或状态变量中连续表征每个参数场。在直接参数测量成本过高或无法获取的各类物理与生物医学问题中，Neptune显著优于现有方法：仅需45个测量点即可实现稳健参数估计，参数估计误差较物理信息神经网络和神经算子等基线方法降低两个数量级，动态响应预测误差降低一个数量级。更重要的是，其具备卓越的物理外推能力，能够在远超训练数据分布范围的场景下实现可靠预测。通过实现可靠且数据高效的参数推断，Neptune有望对工程、医疗健康及其他领域产生广泛变革性影响。