Compositional Generation for Long-Horizon Coupled PDEs

• URL: http://arxiv.org/abs/2510.20141v1
• Date: Thu, 23 Oct 2025 02:35:25 GMT
• Title: Compositional Generation for Long-Horizon Coupled PDEs
• Authors: Somayajulu L. N. Dhulipala, Deep Ray, Nicholas Forman,
• Abstract summary: We study compositional diffusion approaches where diffusion models are only trained on the decoupled PDE data.<n>We investigate whether the compositional strategy can be feasible under long time horizons involving a large number of time steps.<n>We show that compositional diffusion is a viable strategy towards efficient, long-horizon modeling of coupled PDEs.
• Score: 0.764671395172401
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Simulating coupled PDE systems is computationally intensive, and prior efforts have largely focused on training surrogates on the joint (coupled) data, which requires a large amount of data. In the paper, we study compositional diffusion approaches where diffusion models are only trained on the decoupled PDE data and are composed at inference time to recover the coupled field. Specifically, we investigate whether the compositional strategy can be feasible under long time horizons involving a large number of time steps. In addition, we compare a baseline diffusion model with that trained using the v-parameterization strategy. We also introduce a symmetric compositional scheme for the coupled fields based on the Euler scheme. We evaluate on Reaction-Diffusion and modified Burgers with longer time grids, and benchmark against a Fourier Neural Operator trained on coupled data. Despite seeing only decoupled training data, the compositional diffusion models recover coupled trajectories with low error. v-parameterization can improve accuracy over a baseline diffusion model, while the neural operator surrogate remains strongest given that it is trained on the coupled data. These results show that compositional diffusion is a viable strategy towards efficient, long-horizon modeling of coupled PDEs.

Related papers

• GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesis [54.39598154430305]
  We propose a Data-to-Data Flow Matching framework that learns deterministic transformations directly between paired views.<n>PDG-FM constrains flow trajectories using geodesic interpolants derived from probability density metrics of pretrained diffusion models.<n>These results highlight the advantages of incorporating data-dependent geometric regularization into deterministic flow matching for consistent novel view generation.
  arXiv  Detail & Related papers  (2026-03-01T09:30:11Z)
• Decoupled Diffusion Sampling for Inverse Problems on Function Spaces [73.52103661482242]
  Existing plug-and-play diffusion posterior samplers represent physics implicitly through coefficient joint-solution modeling.<n>We propose a physics-aware generative framework in function space for inverse PDE problems.<n>Our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance.
  arXiv  Detail & Related papers  (2026-01-30T18:54:49Z)
• Nonparametric Data Attribution for Diffusion Models [57.820618036556084]
  Data attribution for generative models seeks to quantify the influence of individual training examples on model outputs.<n>We propose a nonparametric attribution method that operates entirely on data, measuring influence via patch-level similarity between generated and training images.
  arXiv  Detail & Related papers  (2025-10-16T03:37:16Z)
• Generative Latent Neural PDE Solver using Flow Matching [8.397730500554047]
  We propose a latent diffusion model for PDE simulation that embeds the PDE state in a lower-dimensional latent space.<n>Our framework uses an autoencoder to map different types of meshes onto a unified structured latent grid, capturing complex geometries.<n> Numerical experiments show that the proposed model outperforms several deterministic baselines in both accuracy and long-term stability.
  arXiv  Detail & Related papers  (2025-03-28T16:44:28Z)
• Towards Theoretical Understandings of Self-Consuming Generative Models [56.84592466204185]
  This paper tackles the emerging challenge of training generative models within a self-consuming loop. We construct a theoretical framework to rigorously evaluate how this training procedure impacts the data distributions learned by future models. We present results for kernel density estimation, delivering nuanced insights such as the impact of mixed data training on error propagation.
  arXiv  Detail & Related papers  (2024-02-19T02:08:09Z)
• Neural variational Data Assimilation with Uncertainty Quantification using SPDE priors [28.804041716140194]
  Recent advances in the deep learning community enables to address the problem through a neural architecture a variational data assimilation framework.<n>In this work we use the theory of Partial Differential Equations (SPDE) and Gaussian Processes (GP) to estimate both space-and time covariance of the state.
  arXiv  Detail & Related papers  (2024-02-02T19:18:12Z)
• PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers [40.097474800631]
  Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Deep neural network based surrogates have gained increased interest.
  arXiv  Detail & Related papers  (2023-08-10T17:53:05Z)
• Training Deep Surrogate Models with Large Scale Online Learning [48.7576911714538]
  Deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. It proposes an open source online training framework for deep surrogate models.
  arXiv  Detail & Related papers  (2023-06-28T12:02:27Z)
• Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data [68.62134204367668]
  This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. The generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution.
  arXiv  Detail & Related papers  (2023-02-14T17:02:35Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• How Much is Enough? A Study on Diffusion Times in Score-based Generative Models [76.76860707897413]
  Current best practice advocates for a large T to ensure that the forward dynamics brings the diffusion sufficiently close to a known and simple noise distribution. We show how an auxiliary model can be used to bridge the gap between the ideal and the simulated forward dynamics, followed by a standard reverse diffusion process.
  arXiv  Detail & Related papers  (2022-06-10T15:09:46Z)
• Transfer Learning on Multi-Fidelity Data [0.0]
  Neural networks (NNs) are often used as surrogates or emulators of partial differential equations (PDEs) that describe the dynamics of complex systems. We rely on multi-fidelity simulations to reduce the cost of data generation for subsequent training of a deep convolutional NN (CNN) using transfer learning. Our numerical experiments demonstrate that a mixture of a comparatively large number of low-fidelity data and smaller numbers of high- and low-fidelity data provides an optimal balance of computational speed-up and prediction accuracy.
  arXiv  Detail & Related papers  (2021-04-29T00:06:19Z)

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.