Spatiotemporal prediction plays an important role in solving natural problems
and processing video frames, especially in weather forecasting and human action
recognition. Recent advances attempt to incorporate prior physical knowledge
into the deep learning framework to estimate the unknown governing partial
differential equations (PDEs), which have shown promising results in
spatiotemporal prediction tasks. However, previous approaches only restrict
neural network architectures or loss functions to acquire physical or PDE
features, which decreases the representative capacity of a neural network.
Meanwhile, the updating process of the physical state cannot be effectively
estimated. To solve the above mentioned problems, this paper proposes a
physical-guided neural network, which utilizes the frequency-enhanced Fourier
module and moment loss to strengthen the model's ability to estimate the
spatiotemporal dynamics. Furthermore, we propose an adaptive second-order
Runge-Kutta method with physical constraints to model the physical states more
precisely. We evaluate our model on both spatiotemporal and video prediction
tasks. The experimental results show that our model outperforms
state-of-the-art methods and performs best in several datasets, with a much
smaller parameter count.

该研究论文提出了一种物理引导的神经网络，利用频率增强的傅里叶模块和矩损失来增强模型估计时空动态的能力，并提出了具有物理约束的自适应二阶龙格 - 库塔方法，以更准确地建模物理状态。实验结果表明，该模型在时空预测和视频预测任务中表现出色，在多个数据集上具有最佳性能，而参数数量却更少。

利用物理引导和频率增强的循环神经网络增强时空预测

Enhanced Spatiotemporal Prediction Using Physical-guided And  Frequency-enhanced Recurrent Neural Networks

In this work, we present an adjoint-based method for discovering the
underlying governing partial differential equations (PDEs) given data. The idea
is to consider a parameterized PDE in a general form, and formulate the
optimization problem that minimizes the error of PDE solution from data. Using
variational calculus, we obtain an evolution equation for the Lagrange
multipliers (adjoint equations) allowing us to compute the gradient of the
objective function with respect to the parameters of PDEs given data in a
straightforward manner. In particular, for a family of parameterized and
nonlinear PDEs, we show how the corresponding adjoint equations can be derived.
Here, we show that given smooth data set, the proposed adjoint method can
recover the true PDE up to machine accuracy. However, in the presence of noise,
the accuracy of the adjoint method becomes comparable to the famous PDE
Functional Identification of Nonlinear Dynamics method known as PDE-FIND (Rudy
et al., 2017). Even though the presented adjoint method relies on
forward/backward solvers, it outperforms PDE-FIND for large data sets thanks to
the analytic expressions for gradients of the cost function with respect to
each PDE parameter.

本研究提出一种基于伴随方法的优化问题，用于从数据中发现潜在的偏微分方程，通过考虑参数化的偏微分方程形式，并最小化 PDE 解与数据之间的误差来计算 PDE 参数的梯度。该方法通过变分计算获取了保正参数的演化方程，可以精确地还原真实的 PDE，尽管在存在噪声的情况下，方法精确度与 PDE-FIND 方法相当。