Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations

• URL: http://arxiv.org/abs/2608.23546v1
• Date: Mon, 24 Aug 2026 17:49:30 GMT
• Title: Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
• Abstract summary: In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs)<n>Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability.<n>For shift-equivariant PDEs, we introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output.
• Score: 0.5586191108738563
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.

Related papers

• Stable by Construction: Variational Latent Markov Operators for Long-Horizon PDE Prediction [12.736876538921626]
  We introduce latent Markov dynamics in which physical states are represented by latent distributions and evolved through probabilistic transitions.<n>We instantiate the framework as the Variational Autoencoding Markov Operator (VAMO), which combines spatially resolved latent fields, structured Gaussian perturbations, and a neural-operator transition.
  arXiv  Detail & Related papers  (2026-09-15T04:28:59Z)
• Local gradient neural operator [12.997669738865484]
  We propose a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems.<n>The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels.<n>We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases.
  arXiv  Detail & Related papers  (2026-09-07T16:53:47Z)
• Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction [17.3214708475049]
  We propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction.<n>GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands.<n>Experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO consistently achieves strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.
  arXiv  Detail & Related papers  (2026-07-31T07:46:35Z)
• Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems [48.470700002941136]
  Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization.<n> physics-informed neural networks (PINNs) have emerged as a flexible alternative.<n>We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems.
  arXiv  Detail & Related papers  (2026-06-10T17:07:09Z)
• Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation [3.918109426744543]
  We study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation.<n>Under smoothness shifts, both models maintain stable performance, with FNO achieving lower error.<n>Under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error.
  arXiv  Detail & Related papers  (2026-05-13T04:53:36Z)
• Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations [0.0]
  We introduce the Shearlet Neural Operator (SNO), a neural operator architecture that replaces the Fourier transform with a shearlet-based representation.<n>SNO consistently improves predictive accuracy and feature fidelity over FNO baselines, with the largest gains observed in anisotropic and discontinuity-dominated settings.
  arXiv  Detail & Related papers  (2026-04-28T03:38:32Z)
• Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations [11.064132774859553]
  Physics-Informed Laplace Neural Operator (PILNO) is a fast surrogate solver for partial differential equations.<n>It embeds physics into training through PDE, boundary condition, and initial condition residuals.<n>PILNO consistently improves accuracy in small-data settings, reduces run-to-run variability across random seeds, and achieves stronger generalization than purely data-driven baselines.
  arXiv  Detail & Related papers  (2026-02-13T08:19:40Z)
• Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations [65.80144621950981]
  We build on Wiener chaos expansions (WCE) to design neural operator (NO) architectures for SPDEs and SDEs.<n>We show that WCE-based neural operators provide a practical and scalable way to learn SDE/SPDE solution operators.
  arXiv  Detail & Related papers  (2026-01-03T00:59:25Z)
• An Inverse Scattering Inspired Fourier Neural Operator for Time-Dependent PDE Learning [0.0]
  We introduce an inverse scattering inspired Fourier Neural Operator (IS-FNO)<n>IS-FNO achieves lower short-term errors and substantially improved long-horizon stability in non-stiff regimes.<n>Overall, this work shows that incorporating physical structure -- particularly reversibility and spectral evolution -- into neural operator design significantly enhances robustness and long-term predictive fidelity for nonlinear PDE dynamics.
  arXiv  Detail & Related papers  (2025-12-22T14:40:13Z)
• 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)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers.<n>We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.<n>Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs [70.51212431290611]
  Partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. We propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-04-13T08:53:41Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Incorporating NODE with Pre-trained Neural Differential Operator for Learning Dynamics [73.77459272878025]
  We propose to enhance the supervised signal in learning dynamics by pre-training a neural differential operator (NDO) NDO is pre-trained on a class of symbolic functions, and it learns the mapping between the trajectory samples of these functions to their derivatives. We provide theoretical guarantee on that the output of NDO can well approximate the ground truth derivatives by proper tuning the complexity of the library.
  arXiv  Detail & Related papers  (2021-06-08T08:04:47Z)

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.