arXiv subjects

Han Wan

Publications and source records attributed to Han Wan.

12 recordsLinked to original sources

arXiv2026

Large language models for partial differential equation workflows

Partial differential equations (PDEs) become actionable in science and engineering not as isolated formulae, but as executable workflows that connect modelling assumptions, governing equations, numerical solvers, diagnostics, and decisions. Large language models (LLMs) are beginning to support such workflows by linking natural language, symbolic mathematics, code, solver outputs, and feedback. Here we examine recent advances in LLM-assisted PDE research across three stages: the discovery and formulation of governing models, the generation and revision of executable numerical solvers, and the use of simulation feedback to support control, design, and optimization. Across these stages, current systems act primarily as workflow-level interfaces. Despite this progress, the field remains limited by the scarcity of high-quality datasets and benchmarks, especially for knowledge discovery and real-world applications, where expert annotation, executable problem construction, and task-level feedback require substantial domain effort. A further challenge is the persistent gap between simulation-based results and real-world scientific and engineering systems, which limits the direct transfer of numerical simulations, control policies, and optimized designs to practical settings. These challenges make LLM-assisted PDE workflows a critical testbed for developing scientific AI systems that can connect language, computation, physical constraints, and real-world decision-making.

cs.AI↗

arXiv2026

Geometry-Aware Neural Optimizer for Shape Optimization and Inversion

Geometry is central to PDE-governed systems, motivating shape optimization and inversion. Classical pipelines conduct costly forward simulation with geometry processing, requiring substantial expert effort. Neural surrogates accelerate forward analysis but do not close the loop because gradients from objectives to geometry are often unavailable. Existing differentiable methods either rely on restrictive parameterizations or unstable latent optimization driven by scalar objectives, limiting interpretability and part-wise control. To address these challenges, we propose Geometry-Aware Neural Optimizer (\textbf{\textsc{GANO}}), an end-to-end differentiable framework that unifies geometry representation, field-level prediction, and automated optimization/inversion in a single latent-space loop. \textsc{GANO} encodes shapes with an auto-decoder and stabilizes latent updates via a denoising mechanism, and a geometry-informed surrogate provides a reliable gradient pathway for geometry updates. Moreover, \textsc{GANO} supports part-wise control through null-space projection and uses remeshing-free projection to accelerate geometry processing. We further prove that denoising induces an implicit Jacobian regularization that reduces decoder sensitivity, yielding controlled deformations. Experiments on three benchmarks spanning 2D Helmholtz, 2D airfoil, and 3D vehicles show state-of-the-art accuracy and stable, controllable updates, achieving up to +55.9% lift-to-drag improvement for airfoils and ~7% drag reduction for vehicles.

cs.LG↗

arXiv2026

PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics

Reconstructing PDE-governed fields from sparse and irregular measurements is challenging due to their ill-posed nature. Deterministic surrogates are trained on dense fields that struggle with limited measurements and uncertainty quantification. Generative models, by learning distributions over spatiotemporal fields, can better handle sparsity and uncertainty. However, existing generative approaches enforce data consistency and PDE constraints simultaneously via sampling-time gradient guidance, resulting in slow and unstable inference. To this end, we propose PerFlow, a Physics-embedded rectified Flow for efficient sparse reconstruction and uncertainty quantification of spatiotemporal dynamics. PerFlow decouples observation conditioning from physics enforcement, performing guidance-free conditioning by feeding observations into rectified-flow dynamics while embedding hard physics via a constraint-preserving projection (e.g., incompressibility or conservation). Theoretically, we establish invariance guarantees to ensure that trajectories remain on the physics-consistent manifold throughout sampling. Experiments on various PDE systems demonstrate competitive reconstruction accuracy with sound physics consistency, while enabling efficient conditional sampling (e.g., 50 steps) and up to 320x faster inference than 2000-step guided diffusion baselines.

cs.LG↗

arXiv2026

Fast and Effective Computation of Generalized Symmetric Matrix Factorization

In this paper, we study a nonconvex, nonsmooth, and non-Lipschitz generalized symmetric matrix factorization model that unifies a broad class of matrix factorization formulations arising in machine learning, image science, engineering, and related areas. We first establish two exactness properties. On the modeling side, we prove an exact penalty property showing that, under suitable conditions, the symmetry-inducing quadratic penalty enforces symmetry whenever the penalty parameter is sufficiently large but finite, thereby exactly recovering the associated symmetric formulation. On the algorithmic side, we introduce an auxiliary-variable splitting formulation and establish an exact relaxation relationship that rigorously links stationary points of the original objective function to those of a relaxed potential function. Building on these exactness properties, we propose an average-type nonmonotone alternating updating method (A-NAUM) based on the relaxed potential function. At each iteration, A-NAUM alternately updates the two factor blocks by (approximately) minimizing the potential function, while the auxiliary block is updated in closed form. To ensure the convergence and enhance practical performance, we further incorporate an average-type nonmonotone line search and show that it is well-defined under mild conditions. Moreover, based on the Kurdyka-{\L}ojasiewicz property and its associated exponent, we establish global convergence of the entire sequence to a stationary point and derive convergence rate results. Finally, numerical experiments on real datasets demonstrate the efficiency of A-NAUM.

math.OC↗

arXiv2025

Stable spectral neural operator for learning stiff PDE systems from limited data

Accurate modeling of spatiotemporal dynamics is crucial to understanding complex phenomena across science and engineering. However, this task faces a fundamental challenge when the governing equations are unknown and observational data are sparse. System stiffness, the coupling of multiple time-scales, further exacerbates this problem and hinders long-term prediction. Existing methods fall short: purely data-driven methods demand massive datasets, whereas physics-aware approaches are constrained by their reliance on known equations and fine-grained time steps. To overcome these limitations, we introduce an equation-free learning framework, namely, the Stable Spectral Neural Operator (SSNO), for modeling stiff partial differential equation (PDE) systems based on limited data. Instead of encoding specific equation terms, SSNO embeds spectrally inspired structures in its architecture, yielding strong inductive biases for learning the underlying physics. It automatically learns local and global spatial interactions in the frequency domain, while handling system stiffness with a robust integrating factor time-stepping scheme. Demonstrated across multiple 2D and 3D benchmarks in Cartesian and spherical geometries, SSNO achieves prediction errors one to two orders of magnitude lower than leading models. Crucially, it shows remarkable data efficiency, requiring only very few (2--5) training trajectories for robust generalization to out-of-distribution conditions. This work offers a robust and generalizable approach to learning stiff spatiotemporal dynamics from limited data without explicit \textit{a priori} knowledge of PDE terms.

physics.comp-ph↗

arXiv2025

L2V-CoT: Cross-Modal Transfer of Chain-of-Thought Reasoning via Latent Intervention

Recently, Chain-of-Thought (CoT) reasoning has significantly enhanced the capabilities of large language models (LLMs), but Vision-Language Models (VLMs) still struggle with multi-step reasoning tasks due to limited multimodal reasoning data. To bridge this gap, researchers have explored methods to transfer CoT reasoning from LLMs to VLMs. However, existing approaches either need high training costs or require architectural alignment. In this paper, we use Linear Artificial Tomography (LAT) to empirically show that LLMs and VLMs share similar low-frequency latent representations of CoT reasoning despite architectural differences. Based on this insight, we propose L2V-CoT, a novel training-free latent intervention approach that transfers CoT reasoning from LLMs to VLMs. L2V-CoT extracts and resamples low-frequency CoT representations from LLMs in the frequency domain, enabling dimension matching and latent injection into VLMs during inference to enhance reasoning capabilities. Extensive experiments demonstrate that our approach consistently outperforms training-free baselines and even surpasses supervised methods.

cs.CL↗

arXiv2025

Towards Affordable, Adaptive and Automatic GNN Training on CPU-GPU Heterogeneous Platforms

Graph Neural Networks (GNNs) have been widely adopted due to their strong performance. However, GNN training often relies on expensive, high-performance computing platforms, limiting accessibility for many tasks. Profiling of representative GNN workloads indicates that substantial efficiency gains are possible on resource-constrained devices by fully exploiting available resources. This paper introduces A3GNN, a framework for affordable, adaptive, and automatic GNN training on heterogeneous CPU-GPU platforms. It improves resource usage through locality-aware sampling and fine-grained parallelism scheduling. Moreover, it leverages reinforcement learning to explore the design space and achieve pareto-optimal trade-offs among throughput, memory footprint, and accuracy. Experiments show that A3GNN can bridge the performance gap, allowing seven Nvidia 2080Ti GPUs to outperform two A100 GPUs by up to 1.8X in throughput with minimal accuracy loss.

cs.DC↗

arXiv2025

OmniFluids: Physics Pre-trained Modeling of Fluid Dynamics

Computational fluid dynamics (CFD) drives progress in numerous scientific and engineering fields, yet high-fidelity simulations remain computationally prohibitive. While machine learning approaches offer computing acceleration, they typically specialize in single physical systems or require extensive training data, hindering their applicability in highly nonlinear and 3D flow scenarios. To overcome these limitations, we propose OmniFluids, a pure physics pre-trained model that captures fundamental fluid dynamics laws and adapts efficiently to diverse downstream tasks with minimal data. We develop a training framework combining physics-only pre-training, coarse-grid operator distillation, and few-shot fine-tuning. This enables OmniFluids to retain broad physics knowledge while delivering fast and accurate predictions. Architecturally, OmniFluids integrates a mixture of operators, a multi-frame decoder, and factorized Fourier layers, seamlessly incorporating physics-based supervision while allowing efficient and scalable modeling of diverse tasks. Extensive tests on a broad range of 2D and 3D benchmarks show that OmniFluids outperforms state-of-the-art AI-driven methods in terms of flow field prediction and turbulence statistics. It delivers 10--100$\times$ speedups over traditional solvers while maintaining a comparable accuracy and accurately identifies unknown physical parameters from sparse, noisy data. This work demonstrates the potential of training a unified CFD solver exclusively from physics knowledge, offering a new approach for efficient and generalizable modeling across complex fluid systems.

physics.flu-dyn↗

arXiv2025

Spectral-inspired Operator Learning with Limited Data and Unknown Physics

Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability. To address these challenges, we propose Spectral-Inspired Neural Operator (SINO), which can model complex systems from just 2-5 trajectories, without requiring explicit PDE terms. Specifically, SINO automatically captures both local and global spatial derivatives from frequency indices, enabling a compact representation of the underlying differential operators in physics-agnostic regimes. To model nonlinear effects, it employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress aliasing. Extensive experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves state-of-the-art performance, with improvements of 1-2 orders of magnitude in accuracy. Particularly, with only 5 training trajectories, SINO outperforms data-driven methods trained on 1000 trajectories and remains predictive on challenging out-of-distribution cases where other methods fail.

cs.LG↗

arXiv2025

PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems

Accurately modeling and forecasting complex systems governed by partial differential equations (PDEs) is crucial in various scientific and engineering domains. However, traditional numerical methods struggle in real-world scenarios due to incomplete or unknown physical laws. Meanwhile, machine learning approaches often fail to generalize effectively when faced with scarce observational data and the challenge of capturing local and global features. To this end, we propose the Physics-encoded Spectral Attention Network (PeSANet), which integrates local and global information to forecast complex systems with limited data and incomplete physical priors. The model consists of two key components: a physics-encoded block that uses hard constraints to approximate local differential operators from limited data, and a spectral-enhanced block that captures long-range global dependencies in the frequency domain. Specifically, we introduce a novel spectral attention mechanism to model inter-spectrum relationships and learn long-range spatial features. Experimental results demonstrate that PeSANet outperforms existing methods across all metrics, particularly in long-term forecasting accuracy, providing a promising solution for simulating complex systems with limited data and incomplete physics.

cs.LG↗

arXiv2025

PIMRL: Physics-Informed Multi-Scale Recurrent Learning for Burst-Sampled Spatiotemporal Dynamics

Deep learning has shown strong potential in modeling complex spatiotemporal dynamics. However, most existing methods depend on densely and uniformly sampled data, which is often unavailable in practice due to sensor and cost limitations. In many real-world settings, such as mobile sensing and physical experiments, data are burst-sampled with short high-frequency segments followed by long gaps, making it difficult to learn accurate dynamics from sparse observations. To address this issue, we propose Physics-Informed Multi-Scale Recurrent Learning (PIMRL), a novel framework specifically designed for burst-sampled spatiotemporal data. PIMRL combines macro-scale latent dynamics inference with micro-scale adaptive refinement guided by incomplete prior information from partial differential equations (PDEs). It further introduces a temporal message-passing mechanism to effectively propagate information across burst intervals. This multi-scale architecture enables PIMRL to model complex systems accurately even under severe data scarcity. We evaluate our approach on five benchmark datasets involving 1D to 3D multi-scale PDEs. The results show that PIMRL consistently outperforms state-of-the-art baselines, achieving substantial improvements and reducing errors by up to 80% in the most challenging settings, which demonstrates the clear advantage of our model. Our work demonstrates the effectiveness of physics-informed recurrent learning for accurate and efficient modeling of sparse spatiotemporal systems.

cs.LG↗

arXiv2023

TinyFormer: Efficient Transformer Design and Deployment on Tiny Devices

Developing deep learning models on tiny devices (e.g. Microcontroller units, MCUs) has attracted much attention in various embedded IoT applications. However, it is challenging to efficiently design and deploy recent advanced models (e.g. transformers) on tiny devices due to their severe hardware resource constraints. In this work, we propose TinyFormer, a framework specifically designed to develop and deploy resource-efficient transformer models on MCUs. TinyFormer consists of SuperNAS, SparseNAS, and SparseEngine. Separately, SuperNAS aims to search for an appropriate supernet from a vast search space. SparseNAS evaluates the best sparse single-path transformer model from the identified supernet. Finally, SparseEngine efficiently deploys the searched sparse models onto MCUs. To the best of our knowledge, SparseEngine is the first deployment framework capable of performing inference of sparse transformer models on MCUs. Evaluation results on the CIFAR-10 dataset demonstrate that TinyFormer can design efficient transformers with an accuracy of 96.1% while adhering to hardware constraints of 1MB storage and 320KB memory. Additionally, TinyFormer achieves significant speedups in sparse inference, up to 12.2x comparing to the CMSIS-NN library. TinyFormer is believed to bring powerful transformers into TinyML scenarios and to greatly expand the scope of deep learning applications

cs.LG↗