fast diffusion solver design

Diffusion models typically require numerous small-step iterations for high-accuracy sampling, and existing solvers exhibit iteration complexity that grows polynomially with both dimensionality and desired precision. This work proposes a novel solver that synergistically combines low-order function approximation with collocation methods, yielding the first diffusion sampler that relies solely on approximate access to the score of the data distribution. By leveraging the effective radius of the support set of the target distribution to control dimensional effects, the method achieves—through theoretical analysis—an iteration complexity that scales only polylogarithmically with the inverse of the accuracy and is independent of the ambient dimension. This represents a significant breakthrough over conventional samplers, which are fundamentally constrained by their joint dependence on dimensionality and precision.

Modeling long-horizon coupled partial differential equation (PDE) systems typically requires large-scale, fully coupled training data—a major bottleneck in data acquisition and scalability. Method: This paper proposes a compositional diffusion modeling framework that operates solely on *decoupled* single-variable trajectory data. It introduces a v-parameterized diffusion model to enhance generation stability and an Euler-type symmetric composition strategy to enforce coordinated evolution and high-fidelity reconstruction of multi-field coupled dynamics. Crucially, no coupled ground-truth labels are required. Contribution/Results: The method significantly reduces data dependency while achieving superior accuracy: on reaction-diffusion and modified Burgers systems, it attains substantially lower long-sequence coupled-field reconstruction error than the Fourier Neural Operator (FNO). Experiments demonstrate that compositional diffusion modeling enables efficient, robust simulation of complex coupled dynamics under strict decoupled-data constraints—establishing both feasibility and state-of-the-art performance.

This work proposes a generative diffusion model based on a spectral representation latent space to address the generation of solutions to parametric partial differential equations (PDEs) under partial observations, unifying both forward and inverse PDE tasks. The method models the joint distribution of PDE parameters and solutions in a compressed, regularized spectral latent space, incorporating physical constraints and posterior sampling via diffusion to assimilate observational data during inference while ensuring solutions reside within the well-posed function class defined by the PDE operator. Each sampling step integrates Adam optimization to enhance numerical stability. Experiments on Poisson, Helmholtz, and incompressible Navier–Stokes equations demonstrate that the approach significantly outperforms existing diffusion-based PDE solvers under sparse observation conditions, achieving notable advances in both accuracy and computational efficiency.

Diffusion models suffer from high inference costs, scaling linearly—O(d) or O(T)—with data dimensionality or time steps, hindering practical deployment. To address this, we propose a block-parallel Picard iteration framework for accelerated sampling. We provide the first rigorous proof that this approach achieves sublinear time complexity—specifically, ~O(poly log d)—thereby breaking the fundamental linear bottleneck. Our method integrates theoretical analysis grounded in the generalized Girsanov theorem with a dual-path design compatible with both stochastic differential equations (SDEs) and probability flow ordinary differential equations (ODEs). This enables scalable, efficient parallelization across large GPU clusters. Experiments on scientific modeling and image generation tasks demonstrate substantial speedups in sampling efficiency—up to orders of magnitude—without compromising sample quality. The framework establishes a new paradigm for high-dimensional diffusion sampling, uniquely combining theoretical guarantees with engineering feasibility.

To address the high training cost and the trade-off between accuracy and generalizability in existing deep learning–based PDE solvers, this paper proposes the first text-to-PDE generation framework. Methodologically: (1) it introduces a novel paradigm wherein natural language directly drives physics-informed simulation; (2) it designs a mesh-agnostic Mesh Autoencoder coupled with full spatiotemporal diffusion modeling to eliminate autoregressive errors; and (3) it incorporates a text-conditioned latent diffusion model, leveraging language as a compact, interpretable control modality. Experiments demonstrate that our method achieves accuracy on par with state-of-the-art neural PDE solvers on uniform and structured grids, while significantly accelerating inference. It supports multiphysics modeling and arbitrary mesh topologies, scales to 3 billion parameters, and exhibits strong cross-physics generalization and engineering practicality.

Diffusion-based generative models have achieved remarkable performance across various domains, yet their practical deployment is often limited by high sampling costs. While prior work focuses on training objectives or individual solvers, the holistic design of sampling, specifically solver selection and scheduling, remains dominated by static heuristics. In this work, we revisit this challenge through a geometric lens, proposing SDM, a principled framework that aligns the numerical solver with the intrinsic properties of the diffusion trajectory. By analyzing the ODE dynamics, we show that efficient low-order solvers suffice in early high-noise stages while higher-order solvers can be progressively deployed to handle the increasing non-linearity of later stages. Furthermore, we formalize the scheduling by introducing a Wasserstein-bounded optimization framework. This method systematically derives adaptive timesteps that explicitly bound the local discretization error, ensuring the sampling process remains faithful to the underlying continuous dynamics. Without requiring additional training or architectural modifications, SDM achieves state-of-the-art performance across standard benchmarks, including an FID of 1.93 on CIFAR-10, 2.41 on FFHQ, and 1.98 on AFHQv2, with a reduced number of function evaluations compared to existing samplers. Our code is available at https://github.com/aiimaginglab/sdm.

Existing probabilistic ODE solvers struggle to simultaneously achieve numerical stability and scalability when applied to stiff, high-dimensional problems. This work proposes two complementary strategies to address this challenge: first, a matrix-free update mechanism leveraging Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to attain linear computational complexity while ensuring numerical stability; second, an iterative re-linearization scheme that reformulates the solver into a fully implicit form, further enhancing stability without compromising scalability. The resulting method constitutes the first probabilistic ODE solver that combines high stability with linear scalability, demonstrating substantial improvements over state-of-the-art approaches across multiple benchmark stiff, high-dimensional ODE systems.

In diffusion sampling, ODE solvers suffer from accumulated gradient estimation errors, while SDE-based methods are highly sensitive to amplified discretization errors when the number of steps is limited. To address this trade-off, we propose AdaSDE—a novel single-step adaptive SDE solver that dynamically modulates error correction strength via a learnable scalar coefficient. This coefficient is estimated by a lightweight distillation module, incurring no additional network overhead and maintaining compatibility with mainstream solvers. Theoretically grounded and empirically validated, AdaSDE achieves state-of-the-art performance with only five sampling steps: FID = 4.18 on CIFAR-10, 8.05 on FFHQ, and 6.96 on LSUN Bedroom. By jointly preserving the computational efficiency of ODE solvers and enhancing robustness against discretization errors inherent to SDEs, AdaSDE significantly improves the speed–quality trade-off in diffusion sampling.

This work addresses the challenge of high-fidelity simulation of high-dimensional chaotic nonlinear dynamical systems, which is computationally prohibitive and difficult to capture with conventional deterministic surrogate models due to their inability to represent intrinsic uncertainties. The authors propose a probabilistic surrogate framework based on diffusion models, integrating a multi-scale graph Transformer with voxel-grid pooling and employing a multi-step autoregressive diffusion training strategy to enable long-term stable predictions. Furthermore, they introduce a retraining-free diffusion posterior sampling mechanism that facilitates uncertainty-aware dynamic sensor placement and efficient data assimilation. Demonstrated on two-dimensional homogeneous isotropic turbulence and backward-facing step flows, the method achieves accurate long-horizon forecasting, adaptive observation layout, and real-time state correction.

This work investigates whether pretrained image editing models can serve as a universal interface for solving diverse physical equations. The approach encodes both inputs and solutions of physical problems as images, incorporates lightweight adapters to embed scalar parameters, and trains the model under a unified architecture using numerical or analytical solutions across multiple equation types—including elliptic, heat, and Navier-Stokes equations. For the first time, it systematically demonstrates that general-purpose generative models can effectively represent both static and dynamic physical mappings, even capturing shocks and unstable phenomena, thereby expanding their applicability in scientific computing. Experiments across more than ten problem classes yield promising results, yet also reveal limitations of image-based representations in handling wide numerical ranges, enforcing constraints, and simulating long-term chaotic dynamics, such as those in the Kuramoto–Sivashinsky equation.