We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR), which models physics signals in a continuous manner and independent of spatial/temporal discretization. The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. PNODE can be inferred by a hypernetwork to reduce the potential difficulties in learning PNODE due to a complex multilayer perceptron (MLP). The framework uses an INR to decode the latent dynamics and reconstruct accurate PDE solutions. Further, a physics-informed loss is also introduced to correct the prediction of unseen parameter instances. Incorporating the physics-informed loss also enables the model to be fine-tuned in an unsupervised manner on unseen PDE parameters. A numerical experiment is performed on a two-dimensional Burgers equation with a large variation of PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(10^3) and ~1% relative error to the ground truth values.

翻译：摘要：本文提出一种新的数据驱动降阶建模方法，用于高效求解多查询问题中的参数化偏微分方程。该工作受隐式神经表示概念的启发，以连续方式建模物理信号，且不依赖于空间/时间离散化。所提出的框架对PDE进行编码，并利用参数化神经常微分方程学习由多个PDE参数表征的潜在动力学。通过超网络推断PNODE可降低因复杂多层感知机带来的学习难度。该框架采用INR解码潜在动力学并重构精确的PDE解。此外，引入物理信息损失修正未观测参数实例的预测。物理信息损失的引入还使得模型能以无监督方式对未观测的PDE参数进行微调。通过对具有大范围PDE参数的二维Burgers方程进行数值实验，我们在高雷诺数下评估所提方法，获得高达O(10^3)的加速比，且与真值相比相对误差约1%。