Physics-informed neural networks have emerged as a transformative framework for addressing operator learning and inverse problems involving the Korteweg–de Vries (KdV) equation for internal solitary waves. By integrating physical constraints with data-driven optimization, PINNs overcome the critical challenges of parameter unmeasurability in the KdV equation for internal solitary waves in two-layer fluid systems. This work addresses two problems: (1) operator learning constructs a mapping from parameters to solutions, enabling wave evolution predictions from unknown parameters. Comparative studies demonstrate prediction errors as low as 10−4 when using 1000 training points. (2) Inverse problem solving leverages sparse and potentially noisy observational data with physics-regularized constraints to invert nonlinear coefficients successfully. Compared to conventional approaches, this end-to-end differentiable paradigm unifies operator learning and inverse problem-solving while overcoming mesh discretization errors and high-dimensional parameter space iteration costs. The method shows effectiveness for internal wave problems in stratified fluids, providing both accurate forward modeling and robust parameter inversion capabilities, even under noise.

We introduce a dual-network physics-informed learning framework designed to solve challenging optimal control problems governed by nonlinear hyperbolic conservation laws. This framework synergistically couples a control variable generator network (CVGN) with an asymptotic-preserving physics-informed neural network (AP-PINN) to jointly discover an optimal control policy and solve the underlying system dynamics. The CVGN parameterizes the control field as a flexible, nonlinear function of space-time coordinates through a neural network. The core innovation lies in the AP-PINN, which addresses the critical challenge of discontinuity in the conservation law solution by solving the corresponding relaxation system instead. This approach transforms the governing equations into a differentiable form amenable to gradient-based optimization. Through a composite loss function, our framework reconciles the optimization objective with physical fidelity without the need for explicit shock tracking. Using the Euler equations as an example of the conservation laws, numerical experiments in one and two dimensions validate the framework's ability to solve complex optimal control problems, including both initial and boundary control scenarios, thereby demonstrating its robustness and versatility.

Land cover dynamics play a critical role in understanding environmental changes, but accurately modeling these dynamics remains a challenge due to the complex interactions between temporal and spatial factors. In this study, we propose a novel temporal–spatial partial differential equation (TS-PDE) modeling method combining sparse regression to uncover the governing equations behind long-term satellite image time series. By integrating temporal and spatial differential terms, the TS-PDE framework captures the intricate interactivity of these factors, overcoming the limitations of traditional pixel-wise prediction methods. Our approach leverages $1\times 1$ convolutional kernels within a convolutional neural network (CNN) solver to approximate derivatives, enabling the discovery of interpretable equations that generalize across temporal–spatial domains. Using MODIS and Planet satellite data, we demonstrate the effectiveness of the TS-PDE method in predicting the value of the normalized difference vegetation index (NDVI) and interpreting the physical significance of the derived equations. The numerical results show that the model achieves good performance, with mean structural similarity index (SSIM) values exceeding 0.82, mean peak signal-to-noise ratio (PSNR) values ranging from 28.5 to 32.8, and mean mean squared error (MSE) values approximating $9\times {10}^{-4}$ for low-resolution MODIS images. For high-resolution Planet images, this study emphasizes the efficacy of TS-PDE in terms of PSNR, SSIM, and MSE metrics, with all datasets exhibiting an average SSIM value of over 0.81, an average PSNR maximum of 30.9, and an average MSE of less than 0.0042. The experimental findings demonstrate the capability of TS-PDE in deriving governing equations and providing effective predictions for the regional-scale dynamics of these time series images. The findings of this study provide potential insights into the mathematical modeling of land cover dynamics.