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Guang Lin

Publications and source records attributed to Guang Lin.

At least 55 records · Page 3Linked to original sources
arXiv2026

pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning

Generalizing across disparate physical laws remains a fundamental challenge for artificial intelligence in science. Existing deep-learning solvers are largely confined to single-equation settings, limiting transfer across physical regimes and inference tasks. Here we introduce pADAM, a unified generative framework that learns a shared probabilistic prior across heterogeneous partial differential equation families. Through a learned joint distribution of system states and, where applicable, physical parameters, pADAM supports forward prediction and inverse inference within a single architecture without retraining. Across benchmarks ranging from scalar diffusion to nonlinear Navier--Stokes equations, pADAM achieves accurate inference even under sparse observations. Combined with conformal prediction, it also provides reliable uncertainty quantification with coverage guarantees. In addition, pADAM performs probabilistic model selection from only two sparse snapshots, identifying governing laws through its learned generative representation. These results highlight the potential of generative multi-physics modeling for unified and uncertainty-aware scientific inference.

cs.LG↗
arXiv2026

Training-Free Global Geometric Association for 4D LiDAR Panoptic Segmentation

Dominant paradigms for 4D LiDAR panoptic segmentation are usually required to train deep neural networks with large superimposed point clouds or design dedicated modules for instance association. However, these approaches perform redundant point processing and consequently become computationally expensive, yet still overlook the rich geometric priors inherently provided by raw point clouds. To this end, we introduce \textsc{Geo-4D}, a simple yet effective training-free framework that unifies spatial and temporal reasoning, enabling holistic LiDAR perception over long time horizons. Specifically, we propose a global geometric association strategy that establishes consistent instance correspondences by estimating an optimal transformation between instance-level point sets. To mitigate instability caused by structural inconsistencies in point cloud observations, we propose a global geometry-aware soft matching mechanism that enforces spatially coherent point-wise correspondences grounded in the spatial distribution of instance point sets. Furthermore, our carefully designed pipeline, which considers three instance types-static, dynamic, and missing-offers computational efficiency and occlusion-aware matching. Our extensive experiments across both SemanticKITTI and nuScenes demonstrate that our method consistently outperforms state-of-the-art approaches, even without additional training or extra point cloud inputs.

cs.CV↗
arXiv2026

Ultra-Fast Language Generation via Discrete Diffusion Divergence Instruct

Fast and high-quality language generation is the holy grail that people pursue in the age of AI. In this work, we introduce Discrete Diffusion Divergence Instruct (DiDi-Instruct), a training-based method that initializes from a pre-trained diffusion large language model (dLLM) and distills a few-step student for fast generation. The model distilled with DiDi-Instruct matches or surpasses its dLLM teacher and the GPT-2 baseline while providing up to 64$\times$ acceleration. The theoretical foundation of DiDi-Instruct is a novel framework based on integral KL-divergence minimization, which leads to a practical training algorithm. We further introduce grouped reward normalization, intermediate-state matching, and the reward-guided ancestral sampler to improve training stability, model coverage, and inference quality. On the OpenWebText benchmark, DiDi-Instruct achieves perplexity ranging from 62.2 (8 NFEs) to 18.4 (128 NFEs), outperforming prior accelerated dLLMs and the GPT-2 baseline. These gains incur a negligible entropy loss (around $1$%) and reduce additional training wall-clock time by more than $20\times$ compared to competing dLLM distillation methods. We further validate the robustness and effectiveness of DiDi-Instruct through extensive ablation studies, model scaling, downstream task evaluations, and unconditional protein sequence generation. In conclusion, DiDi-Instruct enables efficient and effective distillation for language generation in the blink of an eye.

cs.CL↗
arXiv2026

LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning

Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. This makes them difficult to reuse when mechanisms change and it can limit stability under long-horizon rollout. We introduce Lego-like Operator Network (LegONet), a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations. LegONet separates boundary handling from mechanism learning, satisfying boundary conditions by construction. It also separates mechanism learning from time integration, enabling pretrained blocks to be assembled into new solvers without retraining. We also derive a finite-horizon error decomposition that separates block mismatch from splitting error and provides mechanism-level diagnostics for long-horizon predictions. Across ten time-dependent PDEs, LegONet delivers accurate closed-loop rollouts with improved stability under cross-PDE recombination and boundary reconfiguration. More broadly, this modular formulation suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.

math.NA↗
arXiv2026

Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition

AI for science (AI4Science) models often suffer from discretization: learned representations remain tied to the training grid, limiting transfer across resolutions, solvers and applications. We introduce Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator that learns nonlinear, orthogonal basis functions directly in function space and can be integrated in both projection-based reduced order models and operator-learning frameworks such as DeepONet. Neural-POD replaces SVD-derived, resolution-dependent linear modes with continuous, resolution-invariant bases learned via sequential residual minimization, analogous to Gram-Schmidt orthogonalization. The framework supports training under task-specific norms (e.g., $L^2$, $L^1$), improves out-of-distribution generalization to unseen parameter regimes, and captures nonlinear structure in complex systems. Because the learned bases are interpretable and reusable, Neural-POD serves as a general representation module for AI4Science workflows. We demonstrate Neural-POD on Burgers' and Navier-Stokes equations.

physics.comp-ph↗
arXiv2026

Active operator learning with predictive uncertainty quantification for partial differential equations

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. We propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework's uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

cs.LG↗
arXiv2026

Morephy-Net: An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Neural Operator Learning Networks

We propose an evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed operator-learning Networks (Morephy-Net) to solve parametric partial differential equations (PDEs) in noisy data regimes, for both forward prediction and inverse identification. Existing physics-informed neural networks and operator-learning models (e.g., DeepONets and Fourier neural operators) often face three coupled challenges: (i) balancing data/operator and physics residual losses, (ii) maintaining robustness under noisy or sparse observations, and (iii) providing reliable uncertainty quantification. Morephy-Net addresses these issues by integrating: (i) evolutionary multi-objective optimization that treats data/operator and physics residual terms as separate objectives and searches the Pareto front, thereby avoiding ad hoc loss weighting; (ii) replica-exchange stochastic gradient Langevin dynamics to enhance global exploration and stabilize training in non-convex landscapes; and (iii) Bayesian uncertainty quantification obtained from stochastic sampling. We validate Morephy-Net on representative forward and inverse problems, including the one-dimensional Burgers equation and the time-fractional mixed diffusion--wave equation. The results demonstrate consistent improvements in accuracy, noise robustness, and calibrated uncertainty estimates over standard operator-learning baselines.

cs.LG↗
arXiv2026

Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations

Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent PDEs via parameter evolution. The parameter updates are obtained by solving a linear system derived from the governing equation residuals at each time step. However, strong-form evolutionary approaches can yield ill-conditioned linear systems due to pointwise residual discretization, and their computational cost scales unfavorably with the number of training samples. To address these limitations, we propose a weak-form evolutionary Kolmogorov-Arnold Network (KAN) for the scalable and accurate prediction of PDE solutions. We decouple the linear system size from the number of training samples through the weak formulation, leading to improved scalability compared to strong-form approaches. We also rigorously enforce boundary conditions by constructing the trial space with boundary-constrained KANs to satisfy Dirichlet and periodic conditions, and by incorporating derivative boundary conditions directly into the weak formulation for Neumann conditions. In conclusion, the proposed weak-form evolutionary KAN framework provides a stable and scalable approach for PDEs and contributes to scientific machine learning with potential relevance to future engineering applications.

cs.LG↗
arXiv2026

Muon with Spectral Guidance: Efficient Optimization for Scientific Machine Learning

Physics-informed neural networks and neural operators often suffer from severe optimization difficulties caused by ill-conditioned gradients, multi-scale spectral behavior, and stiffness induced by physical constraints. Recently, the Muon optimizer has shown promise by performing orthogonalized updates in the singular-vector basis of the gradient, thereby improving geometric conditioning. However, its unit-singular-value updates may lead to overly aggressive steps and lack explicit stability guarantees when applied to physics-informed learning. In this work, we propose SpecMuon, a spectral-aware optimizer that integrates Muon's orthogonalized geometry with a mode-wise relaxed scalar auxiliary variable (RSAV) mechanism. By decomposing matrix-valued gradients into singular modes and applying RSAV updates individually along dominant spectral directions, SpecMuon adaptively regulates step sizes according to the global loss energy while preserving Muon's scale-balancing properties. This formulation interprets optimization as a multi-mode gradient flow and enables principled control of stiff spectral components. We establish rigorous theoretical properties of SpecMuon, including a modified energy dissipation law, positivity and boundedness of auxiliary variables, and global convergence with a linear rate under the Polyak-Lojasiewicz condition. Numerical experiments on physics-informed neural networks, DeepONets, and fractional PINN-DeepONets demonstrate that SpecMuon achieves faster convergence and improved stability compared with Adam, AdamW, and the original Muon optimizer on benchmark problems such as the one-dimensional Burgers equation and fractional partial differential equations.

cs.LG↗
arXiv2026

Structured Unitary Tensor Network Representations for Circuit-Efficient Quantum Data Encoding

Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose TNQE, a circuit-efficient quantum data encoding framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as $0.04\times$ the depth of amplitude encoding, while naturally scaling to high-resolution images ($256 \times 256$) and demonstrating practical feasibility on real quantum hardware.

quant-ph↗
arXiv2026

Ultra Fast PDE Solving via Physics Guided Few-step Diffusion

Diffusion-based models have demonstrated impressive accuracy and generalization in solving partial differential equations (PDEs). However, they still face significant limitations, such as high sampling costs and insufficient physical consistency, stemming from their many-step iterative sampling mechanism and lack of explicit physics constraints. To address these issues, we propose Phys-Instruct, a novel physics-guided distillation framework which not only (1) compresses a pre-trained diffusion PDE solver into a few-step generator via matching generator and prior diffusion distributions to enable rapid sampling, but also (2) enhances the physics consistency by explicitly injecting PDE knowledge through a PDE distillation guidance. Physic-Instruct is built upon a solid theoretical foundation, leading to a practical physics-constrained training objective that admits tractable gradients. Across five PDE benchmarks, Phys-Instruct achieves orders-of-magnitude faster inference while reducing PDE error by more than 8 times compared to state-of-the-art diffusion baselines. Moreover, the resulting unconditional student model functions as a compact prior, enabling efficient and physically consistent inference for various downstream conditional tasks. Our results indicate that Phys-Instruct is a novel, effective, and efficient framework for ultra-fast PDE solving powered by deep generative models.

cs.LG↗
arXiv2026

Task-tailored Pre-processing: Fair Downstream Supervised Learning

Fairness-aware machine learning has recently attracted various communities to mitigate discrimination against certain societal groups in data-driven tasks. For fair supervised learning, particularly in pre-processing, there have been two main categories: data fairness and task-tailored fairness. The former directly finds an intermediate distribution among the groups, independent of the type of the downstream model, so a learned downstream classification/regression model returns similar predictive scores to individuals inputting the same covariates irrespective of their sensitive attributes. The latter explicitly takes the supervised learning task into account when constructing the pre-processing map. In this work, we study algorithmic fairness for supervised learning and argue that the data fairness approaches impose overly strong regularization from the perspective of the HGR correlation. This motivates us to devise a novel pre-processing approach tailored to supervised learning. We account for the trade-off between fairness and utility in obtaining the pre-processing map. Then we study the behavior of arbitrary downstream supervised models learned on the transformed data to find sufficient conditions to guarantee their fairness improvement and utility preservation. To our knowledge, no prior work in the branch of task-tailored methods has theoretically investigated downstream guarantees when using pre-processed data. We further evaluate our framework through comparison studies based on tabular and image data sets, showing the superiority of our framework which preserves consistent trade-offs among multiple downstream models compared to recent competing models. Particularly for computer vision data, we see our method alters only necessary semantic features related to the central machine learning task to achieve fairness.

cs.LG↗
arXiv2026

Noise estimation of SDE from a single data trajectory

In this paper, we propose a data-driven framework for model discovery of stochastic differential equations (SDEs) from a single trajectory, without requiring the ergodicity or stationary assumption on the underlying continuous process. By combining (stochastic) Taylor expansions with Girsanov transformations, and using the drift function's initial value as input, we construct drift estimators while simultaneously recovering the model noise. This allows us to recover the underlying $\mathbb P$ Brownian motion increments. Building on these estimators, we introduce the first stochastic Sparse Identification of Stochastic Differential Equation (SSISDE) algorithm, capable of identifying the governing SDE dynamics from a single observed trajectory without requiring ergodicity or stationarity. To validate the proposed approach, we conduct numerical experiments with both linear and quadratic drift-diffusion functions. Among these, the Black-Scholes SDE is included as a representative case of a system that does not satisfy ergodicity or stationarity.

q-fin.ST↗
arXiv2026

A POD-DeepONet Framework for Forward and Inverse Design of 2D Photonic Crystals

We develop a reduced-order operator-learning framework for forward and inverse band-structure design of two-dimensional photonic crystals with binary, pixel-based $p4m$-symmetric unit cells. We construct a POD--DeepONet surrogate for the discrete band map along the standard high-symmetry path by coupling a POD trunk extracted from high-fidelity finite-element band snapshots with a neural branch network that predicts reduced coefficients. This architecture yields a compact and differentiable forward model that is tailored to the underlying Bloch eigenvalue discretization. We establish continuity of the discrete band map on the relaxed design space and prove a uniform approximation property of the POD--DeepONet surrogate, leading to a natural decomposition of the total surrogate error into POD truncation and network approximation contributions. Building on this forward surrogate, we formulate two end-to-end neural inverse design procedures, namely dispersion-to-structure and band-gap inverse design, with training objectives that combine data misfit, binarity promotion, and supervised regularization to address the intrinsic non-uniqueness of the inverse mapping and to enable stable gradient-based optimization in the relaxed space. Our numerical results show that the proposed framework achieves accurate forward predictions and produces effective inverse designs on practical high-contrast, pixel-based photonic layouts.

physics.optics↗
arXiv2025

iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filter

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving forward and inverse problems involving partial differential equations (PDEs) by incorporating physical laws into the training process. However, the performance of PINNs is often hindered in real-world scenarios involving noisy observational data and missing physics, particularly in inverse problems. In this work, we propose an iterative multi-objective PINN ensemble Kalman filter (iPINNER) framework that improves the robustness and accuracy of PINNs in both forward and inverse problems by using the \textit{ensemble Kalman filter} and the \textit{non-dominated sorting genetic algorithm} III (NSGA-III). Specifically, NSGA-III is used as a multi-objective optimizer that can generate various ensemble members of PINNs along the optimal Pareto front, while accounting the model uncertainty in the solution space. These ensemble members are then utilized within the EnKF to assimilate noisy observational data. The EnKF's analysis is subsequently used to refine the data loss component for retraining the PINNs, thereby iteratively updating their parameters. The iterative procedure generates improved solutions to the PDEs. The proposed method is tested on two benchmark problems: the one-dimensional viscous Burgers equation and the time-fractional mixed diffusion-wave equation (TFMDWE). The numerical results show it outperforms standard PINNs in handling noisy data and missing physics.

cs.LG↗
arXiv2025

Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data

Parametric PDEs power modern simulation, design, and digital-twin systems, yet their many-query workloads still hinge on repeatedly solving large finite-element systems. Existing operator-learning approaches accelerate this process but often rely on opaque learned trunks, require extensive labeled data, or break down when boundary and source data vary independently from physical parameters. We introduce RB-DeepONet, a hybrid operator-learning framework that fuses reduced-basis (RB) numerical structure with the branch-trunk architecture of DeepONet. The trunk is fixed to a rigorously constructed RB space generated offline via Greedy selection, granting physical interpretability, stability, and certified error control. The branch network predicts only RB coefficients and is trained label-free using a projected variational residual that targets the RB-Galerkin solution. For problems with independently varying loads or boundary conditions, we develop boundary and source modal encodings that compress exogenous data into low-dimensional coordinates while preserving accuracy. Combined with affine or empirical interpolation decompositions, RB-DeepONet achieves a strict offline-online split: all heavy lifting occurs offline, and online evaluation scales only with the RB dimension rather than the full mesh. We provide convergence guarantees separating RB approximation error from statistical learning error, and numerical experiments show that RB-DeepONet attains accuracy competitive with intrusive RB-Galerkin, POD-DeepONet, and FEONet while using dramatically fewer trainable parameters and achieving significant speedups. This establishes RB-DeepONet as an efficient, stable, and interpretable operator learner for large-scale parametric PDEs.

cs.LG↗
arXiv2025

Data-driven Feynman-Kac Discovery with Applications to Prediction and Data Generation

In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law.

q-fin.MF↗
arXiv2025

PO-CKAN:Physics Informed Deep Operator Kolmogorov Arnold Networks with Chunk Rational Structure

We propose PO-CKAN, a physics-informed deep operator framework based on Chunkwise Rational Kolmogorov--Arnold Networks (KANs), for approximating the solution operators of partial differential equations. This framework leverages a Deep Operator Network (DeepONet) architecture that incorporates Chunkwise Rational Kolmogorov-Arnold Network (CKAN) sub-networks for enhanced function approximation. The principles of Physics-Informed Neural Networks (PINNs) are integrated into the operator learning framework to enforce physical consistency. This design enables the efficient learning of physically consistent spatio-temporal solution operators and allows for rapid prediction for parametric time-dependent PDEs with varying inputs (e.g., parameters, initial/boundary conditions) after training. Validated on challenging benchmark problems, PO-CKAN demonstrates accurate operator learning with results closely matching high-fidelity solutions. PO-CKAN adopts a DeepONet-style branch--trunk architecture with its sub-networks instantiated as rational KAN modules, and enforces physical consistency via a PDE residual (PINN-style) loss. On Burgers' equation with $ν=0.01$, PO-CKAN reduces the mean relative $L^2$ error by approximately 48\% compared to PI-DeepONet, and achieves competitive accuracy on the Eikonal and diffusion--reaction benchmarks.

cs.LG↗