Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow

• URL: http://arxiv.org/abs/2605.08915v1
• Date: Sat, 09 May 2026 12:29:10 GMT
• Title: Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow
• Abstract summary: We introduce S-Temporal MeanFlow, which functions as a novel PDE solver learning the finite-interval evolution of physical states.<n>We extend the original MeanFlow constraint from the temporal to thetemporal domain, coupling time evolution with spatial consistency.<n>This yields a unified framework naturally accommodating both time-dependent and stationary PDEs.
• Score: 28.327009081337746
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep learning paradigms, such as PINNs and neural operators, have significantly advanced the solving of PDEs. However, they often struggle to capture the continuous integral nature of physical systems, relying either on pointwise residuals that ignore the integral perspective or on pre-discretized temporal grids. Drawing inspiration from MeanFlow, a continuous-time integrator recently developed to efficiently solve generative ODEs, we introduce Spatio-Temporal MeanFlow, which functions as a novel PDE solver learning the finite-interval evolution of physical states. By substituting the generative velocity field with the physical PDE operator, we transform multi-step numerical integration into an efficient prediction with a freely controllable integration length. Crucially, we extend the original MeanFlow constraint from the temporal to the spatio-temporal domain, coupling time evolution with spatial consistency. This yields a unified framework naturally accommodating both time-dependent and stationary PDEs. Comprehensive experiments on benchmarks demonstrate that our approach achieves superior accuracy and inference efficiency over representative baselines. Furthermore, the proposed integral constraint enables excellent generalization to out-of-distribution initial conditions and varying spatial resolutions.

Related papers

• DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains [0.0]
  Time-dependent partial differential equations (PDEs) with localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs)<n>In this work, a deep adaptive framework for PINNs is extended to treating space and time as a unified domain without any explicit time marching.<n>A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn.
  arXiv  Detail & Related papers  (2026-06-04T15:54:25Z)
• Autoregression-Free Neural Operators for Time-Dependent PDEs [22.734418195465043]
  We propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it.<n>We show that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.
  arXiv  Detail & Related papers  (2026-05-25T04:28:41Z)
• Integral regularization PINNs for evolution equations [4.5305564316166675]
  We propose a novel approach that enhances temporal accuracy by incorporating an integral-based residual term into the loss function.<n>This method divides the entire time interval into smaller sub-intervals and enforces constraints over these sub-intervals, thereby improving the resolution and correlation of temporal dynamics.<n> Numerical experiments on benchmark problems demonstrate that IR-PINNs outperform original PINNs and other state-of-the-art methods in capturing long-time behaviors.
  arXiv  Detail & Related papers  (2025-03-31T05:02:59Z)
• MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation [48.41289705783405]
  We propose a PDE-embedded network with multiscale time stepping (MultiPDENet)<n>In particular, we design a convolutional filter based on the structure of finite difference with a small number of parameters to optimize.<n>A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction.
  arXiv  Detail & Related papers  (2025-01-27T12:15:51Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs)<n>We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations.<n>We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction [49.66486092259376]
  The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. We provide a framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and gradient approximation.
  arXiv  Detail & Related papers  (2023-06-12T16:28:11Z)
• Self-Consistent Velocity Matching of Probability Flows [22.2542921090435]
  We present a discretization-free scalable framework for solving a class of partial differential equations (PDEs) The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent. We use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance.
  arXiv  Detail & Related papers  (2023-01-31T16:17:18Z)
• Semi-supervised Learning of Partial Differential Operators and Dynamical Flows [68.77595310155365]
  We present a novel method that combines a hyper-network solver with a Fourier Neural Operator architecture. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, and three spatial dimensions. The results show that the new method improves the learning accuracy at the time point of supervision point, and is able to interpolate and the solutions to any intermediate time.
  arXiv  Detail & Related papers  (2022-07-28T19:59:14Z)
• Learning to Accelerate Partial Differential Equations via Latent Global Evolution [64.72624347511498]
  Latent Evolution of PDEs (LE-PDE) is a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy.
  arXiv  Detail & Related papers  (2022-06-15T17:31:24Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Long-time integration of parametric evolution equations with physics-informed DeepONets [0.0]
  We introduce an effective framework for learning infinite-dimensional operators that map random initial conditions to associated PDE solutions within a short time interval. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model. This introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations.
  arXiv  Detail & Related papers  (2021-06-09T20:46:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.