New AI methods tackle complex differential equations
ByPulseAugur Editorial·[61 sources]·
Researchers are exploring novel neural network architectures and training methodologies to enhance the solution of complex differential equations. Papers introduce reformulated neural operators that incorporate an auxiliary function dimension for improved embedding evolution, and branched neural rough differential equations that offer a unified framework for stochastic and manifold-valued dynamics. Other work focuses on physics-informed neural operators (PINOs), examining training pipelines for efficiency and robustness, and proposing curvature-aware dynamic precision approaches to balance computational cost and accuracy.
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Advances in neural operator and physics-informed network training offer more efficient and accurate solutions for complex scientific simulations.
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Multiple arXiv papers presenting novel research in neural network architectures and training methods for solving differential equations.
arXiv:2606.06164v1 Announce Type: new Abstract: 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 incorp…
arXiv:2505.11766v4 Announce Type: replace Abstract: Neural Operators (NOs) are powerful architectures for learning mappings between function spaces. While most advances focus on refining kernel parameterizations over the $d$-dimensional physical domain, the evolution of lifted em…
arXiv cs.LG
TIER_1English(EN)·Sebastian Neumayer, Daniel Potts, Fabian Taubert·
arXiv:2606.06046v1 Announce Type: cross Abstract: We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with spa…
arXiv:2606.05272v1 Announce Type: new Abstract: Neural rough differential equations (NRDEs) stay accurate under irregular sampling while taking far fewer integration steps than standard neural differential equations, summarising a finely sampled driver by its log-signature and ad…
arXiv:2604.09361v3 Announce Type: replace Abstract: This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain. The proposed method circum…
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 o…
We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matc…
arXiv cs.AI
TIER_1English(EN)·Yingjie Shao, Ioannis N. Athanasiadis, George van Voorn, Taniya Kapoor·
arXiv:2606.04736v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, recent studies show that PI…
arXiv:2505.15497v3 Announce Type: replace Abstract: Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems. However, in safety-critical contexts, …
arXiv cs.LG
TIER_1English(EN)·Anna Lazareva, Alexander Tarakanov·
arXiv:2606.04420v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses. Their performance is often dominated by the choice of loss weights…
Physics-informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, recent studies show that PINN optimisation is sensitive to numerical precisio…
arXiv cs.LG
TIER_1English(EN)·Yuhan Wu, Jan Willem van Beek, Victorita Dolean, Alexander Heinlein·
arXiv:2605.15806v2 Announce Type: replace Abstract: Neural operators excel as deterministic surrogates, but inevitably collapse to the conditional mean when applied to stochastic PDEs, discarding the variance and tail structure upon which uncertainty quantification depends. Recov…
arXiv:2604.07366v2 Announce Type: replace Abstract: Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein scienc…
arXiv cs.AI
TIER_1English(EN)·Sungwon Kim, Juho Song, Seungmin Shin, Guimok Cho, Sangkook Kim, Chanyoung Park·
arXiv:2606.03260v1 Announce Type: cross Abstract: Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solutio…
arXiv:2606.02623v1 Announce Type: cross Abstract: Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurat…
arXiv cs.AI
TIER_1English(EN)·Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette·
arXiv:2605.13834v2 Announce Type: replace-cross Abstract: In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unl…
arXiv:2606.01122v1 Announce Type: new Abstract: We propose a five-step diagnostic protocol for residual-trained neural HJB-PIDE solvers with control-dependent L\'evy jumps, targeting a general failure mode of neural PDE methods: a learned solution can match headline scalar diagno…
arXiv cs.LG
TIER_1English(EN)·Lennon J. Shikhman, Shane Gilbertie·
arXiv:2606.00937v1 Announce Type: new Abstract: Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even …
arXiv cs.LG
TIER_1English(EN)·Seungchan Ko, Jiyeon Kim, Dongwook Shin·
arXiv:2601.00672v2 Announce Type: replace-cross Abstract: In this paper, we study the finite element operator network (FEONet), an operator-learning method for parametric problems, originally introduced in J. Y. Lee, S. Ko, and Y. Hong, Finite Element Operator Network for Solving…
arXiv cs.LG
TIER_1English(EN)·Yuhan Peng, Junwen Dong, Yuzhi Zeng, Hao Li, Ce Ju, Huitao Feng, Diaaeldin Taha, Anna Wienhard, Kelin Xia·
arXiv:2604.20308v2 Announce Type: replace Abstract: Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix…
arXiv cs.LG
TIER_1English(EN)·Till Muser, Alexandra Spitzer, Matti Lassas, Maarten V. de Hoop, Ivan Dokmani\'c·
arXiv:2603.04430v2 Announce Type: replace Abstract: We introduce Flowers, a neural architecture for learning PDE solution operators built entirely from multihead warps. Aside from pointwise channel mixing and a multiscale scaffold, Flowers use no Fourier multipliers, no dot-produ…
arXiv cs.LG
TIER_1English(EN)·Georgios Is. Detorakis·
arXiv:2408.11266v5 Announce Type: replace Abstract: Deep learning is now common across many scientific fields, including the study of partial differential equations. This article provides a brief, accessible introduction to core deep learning concepts, including neural networks, …
arXiv:2605.31027v1 Announce Type: new Abstract: We propose a novel neural network architecture, termed Multi-Scale Separable Fourier Neural Networks (MS-SFNN), for the accurate and efficient solution of linear and nonlinear high-frequency partial differential equations (PDEs). MS…
arXiv cs.AI
TIER_1English(EN)·Tongfei Chen, Jingying Yang, Linlin Yang, Juan Zhang, Jinhu L\"u, David Doermann, Chunyu Xie, Long He, Tian Wang, Guodong Guo, Baochang Zhang·
arXiv:2603.23977v2 Announce Type: replace-cross Abstract: Deep networks often rely on architectural heuristics to shape representation evolution, limiting their ability to model data governed by intrinsic dynamics. We present the Circuit-inspired High-Order Neural Network (CHONN)…
arXiv cs.LG
TIER_1English(EN)·Gyeonghoon Ko, Juho Lee·
arXiv:2605.31106v1 Announce Type: new Abstract: Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat ker…
arXiv cs.LG
TIER_1English(EN)·Enrico Ballini, Allan Peter Engsig-Karup, Tito Andriollo·
arXiv:2605.31231v1 Announce Type: cross Abstract: We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, wh…
arXiv:2604.06881v2 Announce Type: replace Abstract: Neural operators have emerged as powerful surrogates for dynamical systems due to their grid-invariant properties and computational efficiency. However, Fourier-based variants inherently truncate high-frequency components in spe…
arXiv:2508.11911v2 Announce Type: replace-cross Abstract: We introduce a novel data-driven symplectic induced-order modeling (ROM) framework for high-dimensional Hamiltonian systems that unifies latent-space discovery and dynamics learning within a single, end-to-end neural archi…
We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through funct…
Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challengi…
arXiv:2411.03006v4 Announce Type: replace-cross Abstract: Neural networks with piecewise linear activation functions, such as rectified linear units (ReLU) or maxout, are among the most fundamental models in modern machine learning. We make a step towards proving lower bounds on …
arXiv:2512.21311v2 Announce Type: replace Abstract: Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural rep…
arXiv:2605.29688v1 Announce Type: new Abstract: This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linea…
arXiv:2605.28909v1 Announce Type: new Abstract: We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and…
arXiv:2605.08938v2 Announce Type: replace Abstract: Fourier Neural Operators (FNOs) can greatly accelerate PDE simulation, but they are often used without formal guarantees that they preserve basic physical structure. We show that, once the trained weights and grid are fixed, the…
arXiv cs.LG
TIER_1English(EN)·Chanyoung Kim, Myeonghwan Seong, Yujin Kim, Daniel K. Park, Youngjoon Hong·
arXiv:2605.26631v1 Announce Type: cross Abstract: We propose KO-PDE-IDENT, a data-driven framework for identifying parsimonious partial differential equations (PDEs) with false discovery rate (FDR) control. PDE discovery from noisy observations is often hindered by extreme multic…
arXiv cs.LG
TIER_1English(EN)·Zishuo Lan, Junjie Li, Lei Wang, Jincheng Wang·
arXiv:2602.01941v2 Announce Type: replace-cross Abstract: Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates…
arXiv cs.LG
TIER_1English(EN)·Shan Zhong, George Biros·
arXiv:2605.24876v1 Announce Type: cross Abstract: We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Ne…
arXiv:2605.24041v1 Announce Type: cross Abstract: Neural operators serve as fast, data-driven surrogates for scientific modeling but typically rely on a monolithic, single-pass inference procedure that struggles to resolve high-frequency details, a limitation known as spectral bi…
arXiv cs.AI
TIER_1English(EN)·Jiaquan Zhang, Caiyan Qin, Haoyu Bian, Libin Cai, Yi Lu, Chaoning Zhang, Wei Dong, Yuanfang Guo, Yang Yang, Hen Tao Shen·
arXiv:2605.25057v1 Announce Type: cross Abstract: Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN exp…
arXiv cs.AI
TIER_1English(EN)·Shyam Sankaran, Hanwen Wang, Paris Perdikaris·
arXiv:2605.25949v1 Announce Type: cross Abstract: Neural PDE solvers have followed the scaling trajectory of vision and language, with recent foundation models reaching billions of parameters. We argue that scale is a poor substitute for architectural inductive bias in this domai…
arXiv cs.LG
TIER_1English(EN)·Kyriakos C. Georgiou, Constantinos Siettos, Athanasios N. Yannacopoulos·
arXiv:2507.06038v4 Announce Type: replace-cross Abstract: Building on our previous work on Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to inverse problems for linear and nonlinear elliptic partial differential equations. T…
arXiv:2605.25353v1 Announce Type: new Abstract: Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields.Neural networks are well-suited for PDE parameter estimation due to their …
arXiv:2605.24651v1 Announce Type: cross Abstract: We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi…
Neural PDE solvers have followed the scaling trajectory of vision and language, with recent foundation models reaching billions of parameters. We argue that scale is a poor substitute for architectural inductive bias in this domain: structured priors deliver outsized parameter ef…
Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields.Neural networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space t…
arXiv stat.ML
TIER_1English(EN)·Cornelius Otchere, Michael Shields·
arXiv:2606.06171v1 Announce Type: new Abstract: Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict us…
arXiv:2606.06351v1 Announce Type: new Abstract: Vessel trajectory prediction from Automatic Identification System (AIS) data is essential for maritime situational awareness, yet it remains challenging due to irregular sampling, missing reports, and complex dynamics. Beyond accura…
arXiv stat.ML
TIER_1English(EN)·Anshima Singh, David J. Silvester·
arXiv:2605.08318v2 Announce Type: replace-cross Abstract: We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-d…
Vessel trajectory prediction from Automatic Identification System (AIS) data is essential for maritime situational awareness, yet it remains challenging due to irregular sampling, missing reports, and complex dynamics. Beyond accurate point forecasts, maritime applications also d…
arXiv stat.ML
TIER_1English(EN)·David J. Silvester·
Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-di…
Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify th…
arXiv stat.ML
TIER_1English(EN)·Antonio \'Alvarez-L\'opez, Lorenzo Liverani, Enrique Zuazua·
arXiv:2606.00469v1 Announce Type: cross Abstract: We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit…
We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NO…
<!-- SC_OFF --><div class="md"><p>I built a PINN implementation in Python to solve two problems as part of a physics exam project: the damped harmonic oscillator (2nd-order ODE) and the 1D viscid Burgers' equation (nonlinear PDE). Both forward and inverse problems (to estimate un…