Peiyan Hu

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.

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.

Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Ampère equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a two-branch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems. The code can be found at: https://github.com/Peiyannn/MM-PDE.git.

We present the deep random vortex network (DRVN), a novel physics-informed framework for simulating and inferring the fluid dynamics governed by the incompressible Navier–Stokes equations. Unlike the existing physics-informed neural network (PINN), which embeds physical and geometry information through the residual of equations and boundary data, DRVN automatically embeds this information into neural networks through neural random vortex dynamics equivalent to the Navier–Stokes equation. Specifically, the neural random vortex dynamics motivates a Monte Carlo-based loss function for training neural networks, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVN can efficiently solve Navier–Stokes equations with non-differentiable initial conditions and fractional operators. Furthermore, DRVN naturally embeds the boundary conditions into the kernel function of the neural random vortex dynamics and, thus, does not need additional data to obtain boundary information. We conduct experiments on forward and inverse problems with incompressible Navier–Stokes equations. The proposed method achieves accurate results when simulating and when inferring Navier–Stokes equations. For situations that include singular initial conditions and agnostic boundary data, DRVN significantly outperforms the existing PINN method. Furthermore, compared with the conventional adjoint method when solving inverse problems, DRVN achieves a 2 orders of magnitude improvement for the training time with significantly precise estimates.