Haizhao Yang

In this article, we study a machine‐learning‐based solver for high‐dimensional partial differential equations (PDEs). Computing accurate solutions efficiently for such problems remains challenging because of the curse of dimensionality, which severely limits the scalability of classical numerical methods. Our approach builds on the recently developed finite expression method (FEX), which approximates PDE solutions in a function space generated by finitely many analytic expressions. This framework has been shown to achieve high, and in some cases machine‐level, accuracy with polynomial memory complexity and favorable computational cost. We propose an extension of FEX in which the functional pool is generated by shallow neural network operators whose parameters are initialized using the transferable neural network method TransNet. Numerical experiments suggest that the proposed extension is an effective alternative for solving several high‐dimensional PDEs.

Solving partial differential equations (PDEs) with highly oscillatory solutions on complex domains remains a challenging and important problem. High-frequency oscillations and intricate geometries often result in prohibitively expensive representations for traditional numerical methods and lead to difficult optimization landscapes for machine-learning-based approaches. In this work, we introduce an enhanced finite expression (FEX) method designed to address these challenges with improved accuracy, interpretability, and computational efficiency. The proposed framework incorporates three key innovations: a symbolic spectral composition module that enables FEX to learn and represent multiscale oscillatory behavior; a redesigned linear input layer that significantly expands the expressivity of the model; and an eigenvalue formulation that extends FEX to a new class of problems involving eigenvalue PDEs. Through extensive numerical experiments, we demonstrate that FEX accurately resolves oscillatory PDEs on domains containing multiple holes of varying shapes and sizes. Compared with existing neural-network-based solvers, FEX achieves substantially higher accuracy while yielding interpretable, closed-form solutions that expose the underlying structure of the problem. These advantages, often absent in conventional finite element, finite difference, and black-box neural approaches, highlight FEX as a powerful and transparent framework for solving complex PDEs.

Precipitation products such as Integrated Multi‐satellitE Retrievals have coarse resolution ( km), which limits their application in hydrological modeling and extreme weather analysis. We propose the Wavelet Diffusion Model (WDM), a fast generative framework for high‐quality precipitation downscaling trained on multi‐radar multi‐sensor radar data ( km). WDM employs a conditional diffusion model to learn multiscale precipitation structures in the wavelet domain and to reconstruct high resolution precipitation from low resolution data. Compared with pixel space baselines, our approach produces more realistic fields with fewer artifacts and offers 9× faster inference to kilometer scale resolution. Our results demonstrate that WDM addresses both accuracy and speed, providing a promising method for enhancing hydrological forecasts and other geoscience downscaling tasks.

PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often demand high computational cost and suffer from limited interpretability. We introduce \texttt{AutoNumerics}, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions. Unlike black-box neural solvers, our framework generates transparent solvers grounded in classical numerical analysis. We introduce a coarse-to-fine execution strategy and a residual-based self-verification mechanism. Experiments on 24 canonical and real-world PDE problems demonstrate that \texttt{AutoNumerics} achieves competitive or superior accuracy compared to existing neural and LLM-based baselines, and correctly selects numerical schemes based on PDE structural properties, suggesting its viability as an accessible paradigm for automated PDE solving.

Data scarcity is a fundamental barrier in Electrical Impedance Tomography (EIT), as undersampled Dirichlet-to-Neumann (DtN) measurements can substantially degrade conductivity reconstructions. We address this bottleneck by completing partially observed DtN measurements using a diffusion based generative model. Specifically, we train a conditional diffusion model to learn the distribution of DtN data and to infer full measurement vectors given partial observations. Our approach supports flexible source receiver configurations and can be used as a plug in preprocessing step with off the shelf EIT solvers. Under mild assumptions on the polygon conductivity class, we derive nonasymptotic end to end bounds on the distributional discrepancy between the completed and ground truth DtN measurements. In numerical experiments, we couple the proposed diffusion completion procedure with a deep learning based inverse solver and compare its performance against the same solver with full measurement data. The results show that diffusion completion enables reconstructions comparable to the full data baseline while using only 1% of the measurements. In contrast, standard baselines such as matrix completion require 30% of the measurements to achieve similar reconstruction quality.

Solving symmetric positive semidefinite linear systems is an essential task in many scientific computing problems. While Jacobi-type methods, including the classical Jacobi method and the weighted Jacobi method, exhibit simplicity in their forms and friendliness to parallelization, they are not attractive either because of the potential convergence failure or their slow convergence rate. This paper aims to showcase the possibility of improving classical Jacobi-type methods by employing Nesterov’s acceleration technique that results in an accelerated Jacobi-type method with improved convergence properties. Simultaneously, it preserves the appealing features for parallel implementation. In particular, we show that the proposed method has an [Formula: see text] convergence rate in terms of objective function values of the associated convex quadratic optimization problem, where [Formula: see text] denotes the iteration counter. To further improve the practical performance of the proposed method, we also develop and analyze a restarted variant of the method, which is shown to have an [Formula: see text] convergence rate when the coefficient matrix is positive definite. Furthermore, we conduct appropriate numerical experiments to evaluate the efficiency of the proposed method. Our numerical results demonstrate that the proposed method outperforms the classical Jacobi-type methods and the conjugate gradient method, and shows a comparable performance as the preconditioned conjugate gradient method with a diagonal preconditioner. Finally, we develop a parallel implementation and conduct speed-up tests on some large-scale systems. Our results indicate that the proposed framework is highly scalable.

Deep neural networks (DNNs) have recently emerged as effective tools for approximating solution operators of partial differential equations (PDEs) including evolutionary problems. Classical numerical solvers for such PDEs often face challenges of balancing stability constraints and the high computational cost of iterative solvers. In contrast, DNNs offer a data-driven alternative through direct learning of time-stepping operators to achieve this balancing goal. In this work, we provide a rigorous theoretical framework for analyzing the approximation of these operators using feedforward neural networks (FNNs). We derive explicit error estimates that characterize the dependence of the approximation error on the network architecture -- namely its width and depth -- as well as the number of training samples. Furthermore, we establish Lipschitz continuity properties of time-stepping operators associated with classical numerical schemes and identify low-complexity structures inherent in these operators for several classes of PDEs, including reaction-diffusion equations, parabolic equations with external forcing, and scalar conservation laws. Leveraging these structural insights, we obtain generalization bounds that demonstrate efficient learnability without incurring the curse of dimensionality. Finally, we extend our analysis from single-input operator learning to a general multi-input setting, thereby broadening the applicability of our results.