Physics-informed machine learning has emerged as a promising approach for modeling physical systems. However, real-world applications often face significant challenges due to the limitations of partial observations and inaccuracies in governing partial differential equations (PDEs). In this work, we propose a novel physics-informed machine learning method named Physics-Informed method based on Inaccurate PDEs and Partial Observation (PIPO), to overcome the above two challenges under the real-world scenario, which aims to address the problems of super-resolution and forecasting simultaneously in physical systems characterized by partial observations and inaccurate PDEs. The proposed method is motivated by two key considerations. First, despite the inherent inaccuracy of PDEs, the differential terms (such as diffusion and advection terms) contain valuable information that can effectively reduce the hypothesis space, thereby enhancing the model's generalization capability. Second, while the data only provides partial observations, it offers crucial supervised constraints at the observed points. These constraints not only facilitate model optimization but also help prevent the degeneracy of PDE loss, where the PDE loss could yield multiple solutions. PIPO integrates an interpolator, encoder, forecaster, decoder, and parameters learner, which are jointly optimized using data loss and PDE losses to reconstruct high-resolution states and forecast future states using only partial observation data and inaccurate PDEs. We leverage the proposed PIPO method to address a real-world problem in air pollutant concentration fields and wind fields, specifically PM2.5 transport dynamics, which are governed by the advection-diffusion equation with unknown diffusion coefficients and the source term. The results in super-resolution reconstruction, forecasting, and multi-hour forecasting highlight the effectiveness of PIPO in capturing complex spatial and temporal dynamics despite the limitations of partial observations and inaccurate PDEs.

Aiming to track the maximum power point of photovoltaic power generation, a variety of strategies are proposed. The P&O method is a general and available policy, but traditional P&O strategy cannot track MPP quickly, accurately and stably. A variable step sizes P&O method is proposed to optimize the traditional P&O method. The background and working principle of MPPT and Flyback converter are introduced. The traditional P&O approach is critically considered, and a new variable step sizes P&O approach is presented. The simulation is carried out by Matlab/Simulink, and some curve results are obtained. The results of different sizes of steps and irradiation environments are analyzed. The results present that the variable step sizes P&O strategy could obtain MPP faster at lower oscillation rate at steady-state, and increase the efficiency and output of the renewable power generation based on solar photovoltaic.

In science and engineering, machine learning techniques are increasingly successful in physical systems modeling (predicting future states of physical systems). Effectively integrating PDE loss as a constraint of system transition can improve the model's prediction by overcoming generalization issues due to data scarcity, especially when data acquisition is costly. However, in many real-world scenarios, due to sensor limitations, the data we can obtain is often only partial observation, making the calculation of PDE loss seem to be infeasible, as the PDE loss heavily relies on high-resolution states. We carefully study this problem and propose a novel framework named Re-enable PDE Loss under Partial Observation (RPLPO). The key idea is that although enabling PDE loss to constrain system transition solely is infeasible, we can re-enable PDE loss by reconstructing the learnable high-resolution state and constraining system transition simultaneously. Specifically, RPLPO combines an encoding module for reconstructing learnable high-resolution states with a transition module for predicting future states. The two modules are jointly trained by data and PDE loss. We conduct experiments in various physical systems to demonstrate that RPLPO has significant improvement in generalization, even when observation is sparse, irregular, noisy, and PDE is inaccurate.

Neural operators have demonstrated promise in modeling and controlling systems governed by Partial Differential Equations (PDEs). Beyond PDEs, Stochastic Partial Differential Equations (SPDEs) play a critical role in modeling systems influenced by randomness, with applications in finance, physics, and beyond. However, controlling SPDE-governed systems remains a significant challenge. On the one hand, the regularity of the system's state (which can be intuitively understood as smoothness) deteriorates, making modeling and generalization more challenging. On the other hand, this stochasticity also renders control more unstable and thus less accurate. To address this gap, we propose the Model-Based Closed-Loop Control Algorithm (MB-CC), the first model-based closed-loop control method for SPDEs. MB-CC introduces two key innovations to enhance control robustness and efficiency: a Regularity Feature (RF) block and a closed-loop strategy with an operator-encoded policy network. The RF block, inspired by the regularity structure theory of SPDEs, addresses noise-induced irregularities by transforming the network's input—including the system state and noise-perturbed external forces—into a refined feature space for improved forward prediction. Compared to previous works using regularity features, we introduce a new parameterization, data augmentation, and extend the RF block as a plug-and-play component. Additionally, to achieve closed-loop control, we introduce an operator-encoded policy network to map the current state to optimal control, which integrates physical priors and swiftly makes decisions based on states returned by the environment. We conduct a systematic evaluation of MB-CC on two notable SPDEs, showcasing its effectiveness and efficiency. The ablation studies show its ability to handle stochasticity more effectively.

Modeling flow field evolution accurately is important for numerous natural and engineering applications, such as pollutant dispersion in the ocean and atmosphere, yet remains challenging because of the highly nonlinear, multi-physics, and high-dimensional features of flow systems. While traditional equation-based numerical methods suffer from high computational costs, data-driven neural networks struggle with insufficient data and lack physical explainability. The physics-informed neural operator (PINO) addresses this by combining physics and data losses but faces a fundamental gradient imbalance problem. This work proposes a physics-informed fine-tuned neural operator for high-dimensional flow field modeling that decouples the optimization of physics and data losses. Our method first trains the neural network using data loss and then fine-tunes it with physics loss before inference, enabling the model to adapt to evaluation data while respecting physical constraints. This strategy requires no additional training data and can be applied to fit out-of-distribution (OOD) inputs faced during inference. We validate our method using the shallow water equation and advection–diffusion equation using a convolutional neural operator (CNO) as the base architecture. Experimental results show a 26.4% improvement in single-step prediction accuracy and a reduction in error accumulation for multi-step predictions.