Sifan Wang

Nanxi Chen, Chuanjie Cui, Airong Chen et al.

Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently and robustly remains less well-understood than the training of either data-driven neural operators or physics-informed neural networks (PINNs). To bridge this gap, we examine key components of the PINO training pipeline, including architecture design, optimizer choice, loss balancing, and collocation-point sampling strategy. We study three representative operator backbones, Deep Operator Network (DeepONet), Fourier Neural Operator (FNO), and Continuous Vision Transformer (CViT), across five diverse parametric PDE systems. Our results show that CViT provides consistently strong and stable performance across the considered benchmarks. Beyond architecture, we find that several optimization pathologies previously identified in PINN training naturally arise in PINOs, including gradient conflicts and causal violation. We also find that mitigation algorithms developed for PINNs remain effective in the PINO setting. We further compare physics-informed and data-driven training under different data regimes, revealing that a carefully designed physics-informed training pipeline can match, and in some cases, outperform purely data-driven neural operators. Taken together, these findings provide a systematic empirical understanding of the optimization challenges in PINO training and inform a practical pipeline for efficient and robust physics-informed operator learning. Code and data are available at https://github.com/NanxiiChen/PI-CViT.

Sifan Wang, Shyam Sankaran, Paris Perdikaris

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

Sifan Wang, Shyam Sankaran, Hanwen Wang et al.

Physics-informed neural networks (PINNs) have been popularized as a deep learning framework that can seamlessly synthesize observational data and partial differential equation (PDE) constraints. Their practical effectiveness however can be hampered by training pathologies, but also oftentimes by poor choices made by users who lack deep learning expertise. In this paper we present a series of best practices that can significantly improve the training efficiency and overall accuracy of PINNs. We also put forth a series of challenging benchmark problems that highlight some of the most prominent difficulties in training PINNs, and present comprehensive and fully reproducible ablation studies that demonstrate how different architecture choices and training strategies affect the test accuracy of the resulting models. We show that the methods and guiding principles put forth in this study lead to state-of-the-art results and provide strong baselines that future studies should use for comparison purposes. To this end, we also release a highly optimized library in JAX that can be used to reproduce all results reported in this paper, enable future research studies, as well as facilitate easy adaptation to new use-case scenarios.

Arka Daw, Jie Bu, Sifan Wang et al.

Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing the "failure modes" of PINNs, although a thorough understanding of the connection between PINN failure modes and sampling strategies is missing. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that training PINNs relies on successful "propagation" of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures, characterized by highly imbalanced PDE residual fields. To mitigate propagation failures, we propose a novel Retain-Resample-Release sampling (R3) algorithm that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of R3 sampling to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of R3 sampling and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.

Sifan Wang, Hanwen Wang, Jacob H. Seidman et al.

Continuous neural representations have recently emerged as a powerful and flexible alternative to classical discretized representations of signals. However, training them to capture fine details in multi-scale signals is difficult and computationally expensive. Here we propose random weight factorization as a simple drop-in replacement for parameterizing and initializing conventional linear layers in coordinate-based multi-layer perceptrons (MLPs) that significantly accelerates and improves their training. We show how this factorization alters the underlying loss landscape and effectively enables each neuron in the network to learn using its own self-adaptive learning rate. This not only helps with mitigating spectral bias, but also allows networks to quickly recover from poor initializations and reach better local minima. We demonstrate how random weight factorization can be leveraged to improve the training of neural representations on a variety of tasks, including image regression, shape representation, computed tomography, inverse rendering, solving partial differential equations, and learning operators between function spaces.

Zhiwei Fang, Sifan Wang, Paris Perdikaris

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from $O(N^d)$ to $O(N^{d-1})$, where $d$ is the domain's dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems.

Zhiwei Fang, Sifan Wang, Paris Perdikaris

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date, PINNs have not been successful in simulating multi-scale and singular perturbation problems. In this work, we present a new training paradigm referred to as "gradient boosting" (GB), which significantly enhances the performance of physics informed neural networks (PINNs). Rather than learning the solution of a given PDE using a single neural network directly, our algorithm employs a sequence of neural networks to achieve a superior outcome. This approach allows us to solve problems presenting great challenges for traditional PINNs. Our numerical experiments demonstrate the effectiveness of our algorithm through various benchmarks, including comparisons with finite element methods and PINNs. Furthermore, this work also unlocks the door to employing ensemble learning techniques in PINNs, providing opportunities for further improvement in solving PDEs.

Sifan Wang, Tong-Rui Liu, Shyam Sankaran et al.

Heterogeneous materials, crucial in various engineering applications, exhibit complex multiscale behavior, which challenges the effectiveness of traditional computational methods. In this work, we introduce the Micromechanics Transformer ({\em Micrometer}), an artificial intelligence (AI) framework for predicting the mechanical response of heterogeneous materials, bridging the gap between advanced data-driven methods and complex solid mechanics problems. Trained on a large-scale high-resolution dataset of 2D fiber-reinforced composites, Micrometer can achieve state-of-the-art performance in predicting microscale strain fields across a wide range of microstructures, material properties under any loading conditions and We demonstrate the accuracy and computational efficiency of Micrometer through applications in computational homogenization and multiscale modeling, where Micrometer achieves 1\% error in predicting macroscale stress fields while reducing computational time by up to two orders of magnitude compared to conventional numerical solvers. We further showcase the adaptability of the proposed model through transfer learning experiments on new materials with limited data, highlighting its potential to tackle diverse scenarios in mechanical analysis of solid materials. Our work represents a significant step towards AI-driven innovation in computational solid mechanics, addressing the limitations of traditional numerical methods and paving the way for more efficient simulations of heterogeneous materials across various industrial applications.

Shunyuan Mao, Ruobing Dong, Kwang Moo Yi et al.

We introduce Disk2Planet, a machine learning-based tool to infer key parameters in disk-planet systems from observed protoplanetary disk structures. Disk2Planet takes as input the disk structures in the form of two-dimensional density and velocity maps, and outputs disk and planet properties, that is, the Shakura--Sunyaev viscosity, the disk aspect ratio, the planet--star mass ratio, and the planet's radius and azimuth. We integrate the Covariance Matrix Adaptation Evolution Strategy (CMA--ES), an evolutionary algorithm tailored for complex optimization problems, and the Protoplanetary Disk Operator Network (PPDONet), a neural network designed to predict solutions of disk--planet interactions. Our tool is fully automated and can retrieve parameters in one system in three minutes on an Nvidia A100 graphics processing unit. We empirically demonstrate that our tool achieves percent-level or higher accuracy, and is able to handle missing data and unknown levels of noise.

Sifan Wang, Bowen Li, Yuhan Chen et al.

While physics-informed neural networks (PINNs) have become a popular deep learning framework for tackling forward and inverse problems governed by partial differential equations (PDEs), their performance is known to degrade when larger and deeper neural network architectures are employed. Our study identifies that the root of this counter-intuitive behavior lies in the use of multi-layer perceptron (MLP) architectures with non-suitable initialization schemes, which result in poor trainablity for the network derivatives, and ultimately lead to an unstable minimization of the PDE residual loss. To address this, we introduce Physics-informed Residual Adaptive Networks (PirateNets), a novel architecture that is designed to facilitate stable and efficient training of deep PINN models. PirateNets leverage a novel adaptive residual connection, which allows the networks to be initialized as shallow networks that progressively deepen during training. We also show that the proposed initialization scheme allows us to encode appropriate inductive biases corresponding to a given PDE system into the network architecture. We provide comprehensive empirical evidence showing that PirateNets are easier to optimize and can gain accuracy from considerably increased depth, ultimately achieving state-of-the-art results across various benchmarks. All code and data accompanying this manuscript will be made publicly available at \url{https://github.com/PredictiveIntelligenceLab/jaxpi}.

Amir Noorizadegan, Sifan Wang

The Gaussian scale parameter \(ε\) is central to the behavior of Gaussian Kolmogorov--Arnold Networks (KANs), yet its role in deep edge-based architectures has not been studied systematically. In this paper, we investigate how \(ε\) affects Gaussian KANs through first-layer feature geometry, conditioning, and approximation behavior. Our central observation is that scale selection is governed primarily by the first layer, since it is the only layer constructed directly on the input domain and any loss of distinguishability introduced there cannot be recovered by later layers. From this viewpoint, we analyze the first-layer feature matrix and identify a practical operating interval, \[ ε\in \left[\frac{1}{G-1},\frac{2}{G-1}\right], \] where \(G\) denotes the number of Gaussian centers. For the standard shared-center Gaussian KAN used in current practice, we interpret this interval not as a universal optimality result, but as a stable and effective design rule, and validate it through brute-force sweeps over \(ε\) across function-approximation problems with different collocation densities, grid resolutions, network architectures, and input dimensions, as well as a physics-informed Helmholtz problem. We further show that this range is useful for fixed-scale selection, variable-scale constructions, constrained training of \(ε\), and efficient scale search using early training MSE. Finally, using a matched Chebyshev reference, we show that a properly scaled Gaussian KAN can already be competitive in accuracy relative to another standard KAN basis. In this way, the paper positions scale selection as a practical design principle for Gaussian KANs rather than as an ad hoc hyperparameter choice.

Sifan Wang, Shawn Koohy, Yiping Lu et al.

Physics-informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect solutions despite achieving small residual losses. This failure, we argue, is not merely an optimization difficulty. Rather, it reflects a fundamental weakness of the empirical PDE residual loss, which can admit trivial or spurious solutions during training. From this perspective, we revisit pseudo-time stepping, a technique that has recently shown strong empirical success in PINNs. We show that its main benefit is not simply to ease optimization; instead, when combined with collocation-point resampling, it helps reveal and avoid spurious solutions. At the same time, we find that the effectiveness of pseudo-time stepping depends critically on the choice of step size, which cannot be tuned reliably from the training loss alone. To overcome this limitation, we propose an adaptive pseudo-time stepping strategy that selects the step size from a finite-difference surrogate of the local residual Jacobian, yielding the largest step permitted by local stability without per-problem tuning. Across a diverse set of PDE benchmarks, the proposed method consistently improves both accuracy and robustness. Together, these findings provide a clearer understanding of why PINNs fail and suggest a practical pathway toward more reliable physics-informed learning. All code and data accompanying this manuscript are available at https://github.com/sifanexisted/jaxpi2.

Sifan Wang, Zehao Dou, Tong-Rui Liu et al.

Recent advances in generative modeling -- particularly diffusion models and flow matching -- have achieved remarkable success in synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. Here, we introduce $\textbf{FunDiff}$, a novel framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generate continuous functions evaluable at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, showing that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results show that our method generates physically consistent samples with high fidelity to the target distribution and exhibits robustness to noisy and low-resolution data. Code and datasets are publicly available at https://github.com/sifanexisted/fundiff.

Zhikai Wu, Sifan Wang, Shiyang Zhang et al.

Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress in recent years, enabling efficient approximation of complex spatiotemporal dynamics. However, most existing methods rely on fixed time step sizes during rollout, which limits their ability to adapt to varying temporal complexity and often leads to error accumulation. Here, we propose the Time-Adaptive Transformer with Neural Taylor Expansion (TANTE), a novel operator-learning framework that produces continuous-time predictions with adaptive step sizes. TANTE predicts future states by performing a Taylor expansion at the current state, where neural networks learn both the higher-order temporal derivatives and the local radius of convergence. This allows the model to dynamically adjust its rollout based on the local behavior of the solution, thereby reducing cumulative error and improving computational efficiency. We demonstrate the effectiveness of TANTE across a wide range of PDE benchmarks, achieving superior accuracy and adaptability compared to fixed-step baselines, delivering accuracy gains of 60-80 % and speed-ups of 30-40 % at inference time. The code is publicly available at https://github.com/zwu88/TANTE for transparency and reproducibility.

Ziyu Guo, Renrui Zhang, Hongyu Li et al.

Recent advances in visual generation have increasingly explored the integration of reasoning capabilities. They incorporate textual reasoning, i.e., think, either before (as pre-planning) or after (as post-refinement) the generation process, yet they lack on-the-fly multimodal interaction during the generation itself. In this preliminary study, we introduce Thinking-while-Generating (TwiG), the first interleaved framework that enables co-evolving textual reasoning throughout the visual generation process. As visual content is progressively generating, textual reasoning is interleaved to both guide upcoming local regions and reflect on previously synthesized ones. This dynamic interplay produces more context-aware and semantically rich visual outputs. To unveil the potential of this framework, we investigate three candidate strategies, zero-shot prompting, supervised fine-tuning (SFT) on our curated TwiG-50K dataset, and reinforcement learning (RL) via a customized TwiG-GRPO strategy, each offering unique insights into the dynamics of interleaved reasoning. We hope this work inspires further research into interleaving textual reasoning for enhanced visual generation. Code will be released at: https://github.com/ZiyuGuo99/Thinking-while-Generating.

Amir Noorizadegan, Sifan Wang, Leevan Ling

Kolmogorov-Arnold Networks (KANs) have recently emerged as a promising alternative to traditional Multilayer Perceptrons (MLPs), inspired by the Kolmogorov-Arnold representation theorem. Unlike MLPs, which use fixed activation functions on nodes, KANs employ learnable univariate basis functions on edges, offering enhanced expressivity and interpretability. This review provides a systematic and comprehensive overview of the rapidly expanding KAN landscape, moving beyond simple performance comparisons to offer a structured synthesis of theoretical foundations, architectural variants, and practical implementation strategies. By collecting and categorizing a vast array of open-source implementations, we map the vibrant ecosystem supporting KAN development. We begin by bridging the conceptual gap between KANs and MLPs, establishing their formal equivalence and highlighting the superior parameter efficiency of the KAN formulation. A central theme of our review is the critical role of the basis function; we survey a wide array of choices, including B-splines, Chebyshev and Jacobi polynomials, ReLU compositions, Gaussian RBFs, and Fourier series, and analyze their respective trade-offs in terms of smoothness, locality, and computational cost. We then categorize recent advancements into a clear roadmap, covering techniques for improving accuracy, efficiency, and regularization. Key topics include physics-informed loss design, adaptive sampling, domain decomposition, hybrid architectures, and specialized methods for handling discontinuities. Finally, we provide a practical "Choose-Your-KAN" guide to help practitioners select appropriate architectures, and we conclude by identifying current research gaps. The associated GitHub repository https://github.com/AmirNoori68/kan-review complements this paper and serves as a structured reference for ongoing KAN research.

Shunyuan Mao, Ruobing Dong, Lu Lu et al.

We develop a tool, which we name Protoplanetary Disk Operator Network (PPDONet), that can predict the solution of disk-planet interactions in protoplanetary disks in real-time. We base our tool on Deep Operator Networks (DeepONets), a class of neural networks capable of learning non-linear operators to represent deterministic and stochastic differential equations. With PPDONet we map three scalar parameters in a disk-planet system -- the Shakura \& Sunyaev viscosity $α$, the disk aspect ratio $h_\mathrm{0}$, and the planet-star mass ratio $q$ -- to steady-state solutions of the disk surface density, radial velocity, and azimuthal velocity. We demonstrate the accuracy of the PPDONet solutions using a comprehensive set of tests. Our tool is able to predict the outcome of disk-planet interaction for one system in less than a second on a laptop. A public implementation of PPDONet is available at \url{https://github.com/smao-astro/PPDONet}.

Sifan Wang, Hanwen Wang, Paris Perdikaris

Operator learning techniques have recently emerged as a powerful tool for learning maps between infinite-dimensional Banach spaces. Trained under appropriate constraints, they can also be effective in learning the solution operator of partial differential equations (PDEs) in an entirely self-supervised manner. In this work we analyze the training dynamics of deep operator networks (DeepONets) through the lens of Neural Tangent Kernel (NTK) theory, and reveal a bias that favors the approximation of functions with larger magnitudes. To correct this bias we propose to adaptively re-weight the importance of each training example, and demonstrate how this procedure can effectively balance the magnitude of back-propagated gradients during training via gradient descent. We also propose a novel network architecture that is more resilient to vanishing gradient pathologies. Taken together, our developments provide new insights into the training of DeepONets and consistently improve their predictive accuracy by a factor of 10-50x, demonstrated in the challenging setting of learning PDE solution operators in the absence of paired input-output observations. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/ImprovedDeepONets.}

Sifan Wang, Hanwen Wang, Paris Perdikaris

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency. This is accomplished by leveraging automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. We demonstrate that this simple, yet remarkably effective extension can not only yield a significant improvement in the predictive accuracy of DeepOnets, but also greatly reduce the need for large training data-sets. To this end, a remarkable observation is that physics-informed DeepONets are capable of solving parametric partial differential equations (PDEs) without any paired input-output observations, except for a set of given initial or boundary conditions. We illustrate the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs. Strikingly, a trained physics informed DeepOnet model can predict the solution of $\mathcal{O}(10^3)$ time-dependent PDEs in a fraction of a second -- up to three orders of magnitude faster compared a conventional PDE solver. The data and code accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets}.

Sifan Wang, Hanwen Wang, Paris Perdikaris

Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction-diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at \url{https://github.com/PredictiveIntelligenceLab/MultiscalePINNs}.

Sifan Wang, Xinling Yu, Paris Perdikaris

Physics-informed neural networks (PINNs) have lately received great attention thanks to their flexibility in tackling a wide range of forward and inverse problems involving partial differential equations. However, despite their noticeable empirical success, little is known about how such constrained neural networks behave during their training via gradient descent. More importantly, even less is known about why such models sometimes fail to train at all. In this work, we aim to investigate these questions through the lens of the Neural Tangent Kernel (NTK); a kernel that captures the behavior of fully-connected neural networks in the infinite width limit during training via gradient descent. Specifically, we derive the NTK of PINNs and prove that, under appropriate conditions, it converges to a deterministic kernel that stays constant during training in the infinite-width limit. This allows us to analyze the training dynamics of PINNs through the lens of their limiting NTK and find a remarkable discrepancy in the convergence rate of the different loss components contributing to the total training error. To address this fundamental pathology, we propose a novel gradient descent algorithm that utilizes the eigenvalues of the NTK to adaptively calibrate the convergence rate of the total training error. Finally, we perform a series of numerical experiments to verify the correctness of our theory and the practical effectiveness of the proposed algorithms. The data and code accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/PINNsNTK}.

Sifan Wang, Paris Perdikaris

Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/DeepStefan}.

Sifan Wang, Yujun Teng, Paris Perdikaris

The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine learning with a specific focus on the effectiveness of physics-informed neural networks in predicting outcomes of physical systems and discovering hidden physics from noisy data. We will also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training. To address this limitation we present a learning rate annealing algorithm that utilizes gradient statistics during model training to balance the interplay between different terms in composite loss functions. We also propose a novel neural network architecture that is more resilient to such gradient pathologies. Taken together, our developments provide new insights into the training of constrained neural networks and consistently improve the predictive accuracy of physics-informed neural networks by a factor of 50-100x across a range of problems in computational physics. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/GradientPathologiesPINNs}.

Chun-Wun Cheng, Sifan Wang, Carola-Bibiane Schönlieb et al.

Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76\% over the strongest competing baselines while reducing the number of parameters by 81.98\%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.

Sifan Wang, Ananyae Kumar Bhartari, Bowen Li et al.

Multi-task learning through composite loss functions is fundamental to modern deep learning, yet optimizing competing objectives remains challenging. We present new theoretical and practical approaches for addressing directional conflicts between loss terms, demonstrating their effectiveness in physics-informed neural networks (PINNs) where such conflicts are particularly challenging to resolve. Through theoretical analysis, we demonstrate how these conflicts limit first-order methods and show that second-order optimization naturally resolves them through implicit gradient alignment. We prove that SOAP, a recently proposed quasi-Newton method, efficiently approximates the Hessian preconditioner, enabling breakthrough performance in PINNs: state-of-the-art results on 10 challenging PDE benchmarks, including the first successful application to turbulent flows with Reynolds numbers up to 10,000, with 2-10x accuracy improvements over existing methods. We also introduce a novel gradient alignment score that generalizes cosine similarity to multiple gradients, providing a practical tool for analyzing optimization dynamics. Our findings establish frameworks for understanding and resolving gradient conflicts, with broad implications for optimization beyond scientific computing.

Sifan Wang, Jacob H Seidman, Shyam Sankaran et al.

Operator learning, which aims to approximate maps between infinite-dimensional function spaces, is an important area in scientific machine learning with applications across various physical domains. Here we introduce the Continuous Vision Transformer (CViT), a novel neural operator architecture that leverages advances in computer vision to address challenges in learning complex physical systems. CViT combines a vision transformer encoder, a novel grid-based coordinate embedding, and a query-wise cross-attention mechanism to effectively capture multi-scale dependencies. This design allows for flexible output representations and consistent evaluation at arbitrary resolutions. We demonstrate CViT's effectiveness across a diverse range of partial differential equation (PDE) systems, including fluid dynamics, climate modeling, and reaction-diffusion processes. Our comprehensive experiments show that CViT achieves state-of-the-art performance on multiple benchmarks, often surpassing larger foundation models, even without extensive pretraining and roll-out fine-tuning. Taken together, CViT exhibits robust handling of discontinuous solutions, multi-scale features, and intricate spatio-temporal dynamics. Our contributions can be viewed as a significant step towards adapting advanced computer vision architectures for building more flexible and accurate machine learning models in the physical sciences.

Sifan Wang, Shyam Sankaran, Xiantao Fan et al.

Turbulent fluid flows are among the most computationally demanding problems in science, requiring enormous computational resources that become prohibitive at high flow speeds. Physics-informed neural networks (PINNs) represent a radically different approach that trains neural networks directly from physical equations rather than data, offering the potential for continuous, mesh-free solutions. Here we show that appropriately designed PINNs can successfully simulate fully turbulent flows in both two and three dimensions, directly learning solutions to the fundamental fluid equations without traditional computational grids or training data. Our approach combines several algorithmic innovations including adaptive network architectures, causal training, and advanced optimization methods to overcome the inherent challenges of learning chaotic dynamics. Through rigorous validation on challenging turbulence problems, we demonstrate that PINNs accurately reproduce key flow statistics including energy spectra, kinetic energy, enstrophy, and Reynolds stresses. Our results demonstrate that neural equation solvers can handle complex chaotic systems, opening new possibilities for continuous turbulence modeling that transcends traditional computational limitations.

Junfeng Liao, Sifan Wang, Ye Yuan et al.

Federated Learning (FL) has emerged as an excellent solution for performing deep learning on different data owners without exchanging raw data. However, statistical heterogeneity in FL presents a key challenge, leading to a phenomenon of skewness in local model parameter distributions that researchers have largely overlooked. In this work, we propose the concept of parameter skew to describe the phenomenon that can substantially affect the accuracy of global model parameter estimation. Additionally, we introduce FedSA, an aggregation strategy to obtain a high-quality global model, to address the implication from parameter skew. Specifically, we categorize parameters into high-dispersion and low-dispersion groups based on the coefficient of variation. For high-dispersion parameters, Micro-Classes (MIC) and Macro-Classes (MAC) represent the dispersion at the micro and macro levels, respectively, forming the foundation of FedSA. To evaluate the effectiveness of FedSA, we conduct extensive experiments with different FL algorithms on three computer vision datasets. FedSA outperforms eight state-of-the-art baselines by about 4.7% in test accuracy.

Nanxi Chen, Chuanjie Cui, Rujin Ma et al.

Physics-informed neural networks have shown significant potential in solving partial differential equations (PDEs) across diverse scientific fields. However, their performance often deteriorates when addressing PDEs with intricate and strongly coupled solutions. In this work, we present a novel Sharp-PINN framework to tackle complex phase field corrosion problems. Instead of minimizing all governing PDE residuals simultaneously, the Sharp-PINNs introduce a staggered training scheme that alternately minimizes the residuals of Allen-Cahn and Cahn-Hilliard equations, which govern the corrosion system. To further enhance its efficiency and accuracy, we design an advanced neural network architecture that integrates random Fourier features as coordinate embeddings, employs a modified multi-layer perceptron as the primary backbone, and enforces hard constraints in the output layer. This framework is benchmarked through simulations of corrosion problems with multiple pits, where the staggered training scheme and network architecture significantly improve both the efficiency and accuracy of PINNs. Moreover, in three-dimensional cases, our approach is 5-10 times faster than traditional finite element methods while maintaining competitive accuracy, demonstrating its potential for real-world engineering applications in corrosion prediction.

Nanxi Chen, Sifan Wang, Rujin Ma et al.

Physics-informed neural networks (PINNs) represent a new paradigm for solving partial differential equations (PDEs) by integrating physical laws into the learning process of neural networks. However, despite their foundational role, the hidden irreversibility implied by the Second Law of Thermodynamics is often neglected during training, leading to unphysical solutions or even training failures in conventional PINNs. In this paper, we identify this critical gap and introduce a simple, generalized, yet robust irreversibility-regularized strategy that enforces hidden physical laws as soft constraints during training. This approach ensures that the learned solutions consistently respect the intrinsic one-way nature of irreversible physical processes. Across a wide range of benchmarks spanning traveling wave propagation, steady combustion, ice melting, corrosion evolution, and crack propagation, we demonstrate that our regularization scheme reduces predictive errors by more than an order of magnitude, while requiring only minimal modification to existing PINN frameworks. We believe that the proposed framework is broadly applicable to a wide class of PDE-governed physical systems and will have significant impact within the scientific machine learning community.

Fei Ren, Sifan Wang, Pei-Zhi Zhuang et al.

Conventional physics-informed extreme learning machine (PIELM) often faces challenges in solving partial differential equations (PDEs) involving high-frequency and variable-frequency behaviors. To address these challenges, we propose a general Fourier feature physics-informed extreme learning machine (GFF-PIELM). We demonstrate that directly concatenating multiple Fourier feature mappings (FFMs) and an extreme learning machine (ELM) network makes it difficult to determine frequency-related hyperparameters. Fortunately, we find an alternative to establish the GFF-PIELM in three main steps. First, we integrate a variation of FFM into ELM as the Fourier-based activation function, so there is still one hidden layer in the GFF-PIELM framework. Second, we assign a set of frequency coefficients to the hidden neurons, which enables ELM network to capture diverse frequency components of target solutions. Finally, we develop an innovative, straightforward initialization method for these hyperparameters by monitoring the distribution of ELM output weights. GFF-PIELM not only retains the high accuracy, efficiency, and simplicity of the PIELM framework but also inherits the ability of FFMs to effectively handle high-frequency problems. We carry out five case studies with a total of ten numerical examples to highlight the feasibility and validity of the proposed GFF-PIELM, involving high frequency, variable frequency, multi-scale behaviour, irregular boundary and inverse problems. Compared to conventional PIELM, the GFF-PIELM approach significantly improves predictive accuracy without additional cost in training time and architecture complexity. Our results confirm that that PIELM can be extended to solve high-frequency and variable-frequency PDEs with high accuracy, and our initialization strategy may further inspire advances in other physics-informed machine learning (PIML) frameworks.

Sifan Wang, Zhikai Wu, David van Dijk et al.

Inverse problems governed by partial differential equations (PDEs) are crucial in science and engineering. They are particularly challenging due to ill-posedness, data sparsity, and the added complexity of irregular geometries. Classical PDE-constrained optimization methods are computationally expensive, especially when repeated posterior sampling is required. Learning-based approaches improve efficiency and scalability, yet most are designed for regular domains or focus on forward modeling. Here, we introduce {\em GeoFunFlow}, a geometric diffusion model framework for inverse problems on complex geometries. GeoFunFlow combines a novel geometric function autoencoder (GeoFAE) and a latent diffusion model trained via rectified flow. GeoFAE employs a Perceiver module to process unstructured meshes of varying sizes and produces continuous reconstructions of physical fields, while the diffusion model enables posterior sampling from sparse and noisy data. Across five benchmarks, GeoFunFlow achieves state-of-the-art reconstruction accuracy over complex geometries, provides calibrated uncertainty quantification, and delivers efficient inference compared to operator-learning and diffusion model baselines.

Shunyuan Mao, Weiqi Wang, Sifan Wang et al.

Accretion disks are ubiquitous in astrophysics, appearing in diverse environments from planet-forming systems to X-ray binaries and active galactic nuclei. Traditionally, modeling their dynamics requires computationally intensive (magneto)hydrodynamic simulations. Recently, Physics-Informed Neural Networks (PINNs) have emerged as a promising alternative. This approach trains neural networks directly on physical laws without requiring data. We for the first time demonstrate PINNs for solving the two-dimensional, time-dependent hydrodynamics of non-self-gravitating accretion disks. Our models provide solutions at arbitrary times and locations within the training domain, and successfully reproduce key physical phenomena, including the excitation and propagation of spiral density waves and gap formation from disk-companion interactions. Notably, the boundary-free approach enabled by PINNs naturally eliminates the spurious wave reflections at disk edges, which are challenging to suppress in numerical simulations. These results highlight how advanced machine learning techniques can enable physics-driven, data-free modeling of complex astrophysical systems, potentially offering an alternative to traditional numerical simulations in the future.

Sifan Wang, Mohamed Aziz Bhouri, Paris Perdikaris

Design and optimal control problems are among the fundamental, ubiquitous tasks we face in science and engineering. In both cases, we aim to represent and optimize an unknown (black-box) function that associates a performance/outcome to a set of controllable variables through an experiment. In cases where the experimental dynamics can be described by partial differential equations (PDEs), such problems can be mathematically translated into PDE-constrained optimization tasks, which quickly become intractable as the number of control variables and the cost of experiments increases. In this work we leverage physics-informed deep operator networks (DeepONets) -- a self-supervised framework for learning the solution operator of parametric PDEs -- to build fast and differentiable surrogates for rapidly solving PDE-constrained optimization problems, even in the absence of any paired input-output training data. The effectiveness of the proposed framework will be demonstrated across different applications involving continuous functions as control or design variables, including time-dependent optimal control of heat transfer, and drag minimization of obstacles in Stokes flow. In all cases, we observe that DeepONets can minimize high-dimensional cost functionals in a matter of seconds, yielding a significant speed up compared to traditional adjoint PDE solvers that are typically costly and limited to relatively low-dimensional control/design parametrizations.

Sifan Wang, Paris Perdikaris

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to introduce new effective ways of simulating PDEs, however existing approaches are not able to reliably return stable and accurate predictions across long temporal horizons. We aim to address this challenge by introducing an effective framework for learning infinite-dimensional operators that map random initial conditions to associated PDE solutions within a short time interval. Such latent operators can be parametrized by deep neural networks that are trained in an entirely self-supervised manner without requiring any paired input-output observations. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model using each prediction as the initial condition for the next evaluation step. This introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations for a wide range of parametric ODE and PDE systems, from wave propagation, to reaction-diffusion dynamics and stiff chemical kinetics, all at a fraction of the computational cost needed by classical numerical solvers.

Dawei Li, Yan Cao, Guoliang Shi et al.

Automatic leaf segmentation, as well as identification and classification methods that built upon it, are able to provide immediate monitoring for plant growth status to guarantee the output. Although 3D plant point clouds contain abundant phenotypic features, plant leaves are usually distributed in clusters and are sometimes seriously overlapped in the canopy. Therefore, it is still a big challenge to automatically segment each individual leaf from a highly crowded plant canopy in 3D for plant phenotyping purposes. In this work, we propose an overlapping-free individual leaf segmentation method for plant point clouds using the 3D filtering and facet region growing. In order to separate leaves with different overlapping situations, we develop a new 3D joint filtering operator, which integrates a Radius-based Outlier Filter (RBOF) and a Surface Boundary Filter (SBF) to help to separate occluded leaves. By introducing the facet over-segmentation and facet-based region growing, the noise in segmentation is suppressed and labeled leaf centers can expand to their whole leaves, respectively. Our method can work on point clouds generated from three types of 3D imaging platforms, and also suitable for different kinds of plant species. In experiments, it obtains a point-level cover rate of 97% for Epipremnum aureum, 99% for Monstera deliciosa, 99% for Calathea makoyana, and 87% for Hedera nepalensis sample plants. At the leaf level, our method reaches an average Recall at 100.00%, a Precision at 99.33%, and an average F-measure at 99.66%, respectively. The proposed method can also facilitate the automatic traits estimation of each single leaf (such as the leaf area, length, and width), which has potential to become a highly effective tool for plant research and agricultural engineering.

Dawei Li, Siyuan Yan, Xin Cai et al.

Change detection is a fundamental task in computer vision. Despite significant advances have been made, most of the change detection methods fail to work well in challenging scenes due to ubiquitous noise and interferences. Nowadays, post-processing methods (e.g. MRF, and CRF) aiming to enhance the binary change detection results still fall short of the requirements on universality for distinctive scenes, applicability for different types of detection methods, accuracy, and real-time performance. Inspired by the nature of image filtering, which separates noise from pixel observations and recovers the real structure of patches, we consider utilizing image filters to enhance the detection masks. In this paper, we present an integrated filter which comprises a weighted local guided image filter and a weighted spatiotemporal tree filter. The spatiotemporal tree filter leverages the global spatiotemporal information of adjacent video frames and meanwhile the guided filter carries out local window filtering of pixels, for enhancing the coarse change detection masks. The main contributions are three: (i) the proposed filter can make full use of the information of the same object in consecutive frames to improve its current detection mask by computations on a spatiotemporal minimum spanning tree; (ii) the integrated filter possesses both advantages of local filtering and global filtering; it not only has good edge-preserving property but also can handle heavily textured and colorful foreground regions; and (iii) Unlike some popular enhancement methods (MRF, and CRF) that require either a priori background probabilities or a posteriori foreground probabilities for every pixel to improve the coarse detection masks, our method is a versatile enhancement filter that can be applied after many different types of change detection methods, and is particularly suitable for video sequences.