arXiv subjects

Wuzhe Xu

Publications and source records attributed to Wuzhe Xu.

10 recordsLinked to original sources

arXiv2025

Self-Supervised Amortized Neural Operators for Optimal Control: Scaling Laws and Applications

Optimal control provides a principled framework for transforming dynamical system models into intelligent decision-making, yet classical computational approaches are often too expensive for real-time deployment in dynamic or uncertain environments. In this work, we propose a method based on self-supervised neural operators for open-loop optimal control problems. It offers a new paradigm by directly approximating the mapping from system conditions to optimal control strategies, enabling instantaneous inference across diverse scenarios once trained. We further extend this framework to more complex settings, including dynamic or partially observed environments, by integrating the learned solution operator with Model Predictive Control (MPC). This yields a solution-operator learning method for closed-loop control, in which the learned operator supplies rapid predictions that replace the potentially time-consuming optimization step in conventional MPC. This acceleration comes with a quantifiable price to pay. Theoretically, we derive scaling laws that relate generalization error and sample/model complexity to the intrinsic dimension of the problem and the regularity of the optimal control function. Numerically, case studies show efficient, accurate real-time performance in low-intrinsic-dimension regimes, while accuracy degrades as problem complexity increases. Together, these results provide a balanced perspective: neural operators are a powerful novel tool for high-performance control when hidden low-dimensional structure can be exploited, yet they remain fundamentally constrained by the intrinsic dimensional complexity in more challenging settings.

math.OC↗

arXiv2025

In-Context Learning of Linear Systems: Generalization Theory and Applications to Operator Learning

We study theoretical guarantees for solving linear systems in-context using a linear transformer architecture. For in-domain generalization, we provide neural scaling laws that bound the generalization error in terms of the number of tasks and sizes of samples used in training and inference. For out-of-domain generalization, we find that the behavior of trained transformers under task distribution shifts depends crucially on the distribution of the tasks seen during training. We introduce a novel notion of task diversity and show that it defines a necessary and sufficient condition for pre-trained transformers generalize under task distribution shifts. We also explore applications of learning linear systems in-context, such as to in-context operator learning for PDEs. Finally, we provide some numerical experiments to validate the established theory.

cs.LG↗

arXiv2025

Improving Data Fidelity via Diffusion Model-based Correction and Super-Resolution

We propose a unified diffusion model-based correction and super-resolution method to enhance the fidelity and resolution of diverse low-quality data through a two-step pipeline. First, the correction step employs a novel enhanced stochastic differential editing technique based on an imbalanced perturbation and denoising process, ensuring robust and effective bias correction at the low-resolution level. The robustness and effectiveness of this approach are validated theoretically and experimentally. Next, the super-resolution step leverages cascaded conditional diffusion models to iteratively refine the corrected data to high-resolution. Numerical experiments on three PDE problems and a climate dataset demonstrate that the proposed method effectively enhances low-fidelity, low-resolution data by correcting numerical errors and noise while simultaneously improving resolution to recover fine-scale structures.

math.NA↗

arXiv2025

Diffusion-based Models for Unpaired Super-resolution in Fluid Dynamics

High-fidelity, high-resolution numerical simulations are crucial for studying complex multiscale phenomena in fluid dynamics, such as turbulent flows and ocean waves. However, direct numerical simulations with high-resolution solvers are computationally prohibitive. As an alternative, super-resolution techniques enable the enhancement of low-fidelity, low-resolution simulations. However, traditional super-resolution approaches rely on paired low-fidelity, low-resolution and high-fidelity, high-resolution datasets for training, which are often impossible to acquire in complex flow systems. To address this challenge, we propose a novel two-step approach that eliminates the need for paired datasets. First, we perform unpaired domain translation at the low-resolution level using an Enhanced Denoising Diffusion Implicit Bridge. This process transforms low-fidelity, low-resolution inputs into high-fidelity, low-resolution outputs, and we provide a theoretical analysis to highlight the advantages of this enhanced diffusion-based approach. Second, we employ the cascaded Super-Resolution via Repeated Refinement model to upscale the high-fidelity, low-resolution prediction to the high-resolution result. We demonstrate the effectiveness of our approach across three fluid dynamics problems. Moreover, by incorporating a neural operator to learn system dynamics, our method can be extended to improve evolutionary simulations of low-fidelity, low-resolution data.

math.NA↗

arXiv2024

Generative downscaling of PDE solvers with physics-guided diffusion models

Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities of the problems, including nonlinearity and multiscale phenomena. To speed up large-scale computations, a process known as downscaling is employed, which generates high-fidelity approximate solutions from their low-fidelity counterparts. In this paper, we propose a novel Physics-Guided Diffusion Model (PGDM) for downscaling. Our model, initially trained on a dataset comprising low-and-high-fidelity paired solutions across coarse and fine scales, generates new high-fidelity approximations from any new low-fidelity inputs. These outputs are subsequently refined through fine-tuning, aimed at minimizing the physical discrepancies as defined by the discretized PDEs at the finer scale. We evaluate and benchmark our model's performance against other downscaling baselines in three categories of nonlinear PDEs. Our numerical experiments demonstrate that our model not only outperforms the baselines but also achieves a computational acceleration exceeding tenfold, while maintaining the same level of accuracy as the conventional fine-scale solvers.

math.NA↗

arXiv2024

To blow-up or not to blow-up for a granular kinetic equation

A simplified kinetic description of rapid granular media leads to a nonlocal Vlasov-type equation with a convolution integral operator that is of the same form as the continuity equations for aggregation-diffusion macroscopic dynamics. While the singular behavior of these nonlinear continuity equations is well studied in the literature, the extension to the corresponding granular kinetic equation is highly nontrivial. The main question is whether the singularity formed in velocity direction will be enhanced or mitigated by the shear in phase space due to free transport. We present a preliminary study through a meticulous numerical investigation and heuristic arguments. We have numerically developed a structure-preserving method with adaptive mesh refinement that can effectively capture potential blow-up behavior in the solution for granular kinetic equations. We have analytically constructed a finite-time blow-up infinite mass solution and discussed how this can provide insights into the finite mass scenario.

math.NA↗

arXiv2022

Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations

Deep operator network (DeepONet) has demonstrated great success in various learning tasks, including learning solution operators of partial differential equations. In particular, it provides an efficient approach to predict the evolution equations in a finite time horizon. Nevertheless, the vanilla DeepONet suffers from the issue of stability degradation in the long-time prediction. This paper proposes a {\em transfer-learning} aided DeepONet to enhance the stability. Our idea is to use transfer learning to sequentially update the DeepONets as the surrogates for propagators learned in different time frames. The evolving DeepONets can better track the varying complexities of the evolution equations, while only need to be updated by efficient training of a tiny fraction of the operator networks. Through systematic experiments, we show that the proposed method not only improves the long-time accuracy of DeepONet while maintaining similar computational cost but also substantially reduces the sample size of the training set.

cs.LG↗

arXiv2022

Solving multiscale steady radiative transfer equation using neural networks with uniform stability

This paper concerns solving the steady radiative transfer equation with diffusive scaling, using the physics informed neural networks (PINNs). The idea of PINNs is to minimize a least-square loss function, that consists of the residual from the governing equation, the mismatch from the boundary conditions, and other physical constraints such as conservation. It is advantageous of being flexible and easy to execute, and brings the potential for high dimensional problems. Nevertheless, due the presence of small scales, the vanilla PINNs can be extremely unstable for solving multiscale steady transfer equations. In this paper, we propose a new formulation of the loss based on the macro-micro decomposition. We prove that, the new loss function is uniformly stable with respect to the small Knudsen number in the sense that the $L^2$-error of the neural network solution is uniformly controlled by the loss. When the boundary condition is an-isotropic, a boundary layer emerges in the diffusion limit and therefore brings an additional difficulty in training the neural network. To resolve this issue, we include a boundary layer corrector that carries over the sharp transition part of the solution and leaves the rest easy to be approximated. The effectiveness of the new methodology is demonstrated in extensive numerical examples.

math.NA↗

arXiv2021

An asymptotic preserving scheme for Lévy-Fokker-Planck equation with fractional diffusion limit

In this paper, we develop a numerical method for the Lévy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold nonlocality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other arises from long-time/small mean-free-path scaling, which introduces stiffness to the equation. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply the Chebyshev polynomial based pseudo-spectral method. To treat the multiple scales, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically. Finally, the efficiency and accuracy of our scheme are illustrated by a suite of numerical examples.

math.NA↗

arXiv2020

Variational Asymptotic Preserving Scheme for the Vlasov-Poisson-Fokker-Planck System

We design a variational asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system with the high field scaling, which describes the Brownian motion of a large system of particles in a surrounding bath. Our scheme builds on an implicit-explicit framework, wherein the stiff terms coming from the collision and field effects are solved implicitly while the convection terms are solved explicitly. To treat the implicit part, we propose a variational approach by viewing it as a Wasserstein gradient flow of the relative entropy, and solve it via a proximal quasi-Newton method. In so doing we get positivity and asymptotic preservation for free. The method is also massively parallelizable and thus suitable for high dimensional problems. We further show that the convergence of our implicit solver is uniform across different scales. A suite of numerical examples are presented at the end to validate the performance of the proposed scheme.

math.NA↗