In this paper, we present a patch-based representation of surfaces, PolyFit, which is obtained by fitting jet functions locally on surface patches. Such a representation can be learned efficiently in a supervised fashion from both analytic functions and real data. Once learned, it can be generalized to various types of surfaces. Using PolyFit, the surfaces can be efficiently deformed by updating a compact set of jet coefficients rather than optimizing per-vertex degrees of freedom for many downstream tasks in computer vision and graphics. We demonstrate the capabilities of our proposed methodologies with two applications: 1) Shape-from-template (SfT): where the goal is to deform the input 3D template of an object as seen in image/video. Using PolyFit, we adopt test-time optimization that delivers competitive accuracy while being markedly faster than offline physics-based solvers, and outperforms recent physicsguided neural simulators in accuracy at modest additional runtime. 2) Garment draping. We train a self-supervised, mesh- and garment-agnostic model that generalizes across resolutions and garment types, delivering up to an order-ofmagnitude faster inference than strong baselines.

We present FNOpt, a self-supervised cloth simulation framework that formulates time integration as an optimization problem and trains a resolution-agnostic neural optimizer parameterized by a Fourier neural operator (FNO). Prior neural simulators often rely on extensive ground truth data or sacrifice fine-scale detail, and generalize poorly across resolutions and motion patterns. In contrast, FNOpt learns to simulate physically plausible cloth dynamics and achieves stable and accurate rollouts across diverse mesh resolutions and motion patterns without retraining. Trained only on a coarse grid with physics-based losses, FNOpt generalizes to finer resolutions, capturing fine-scale wrinkles and preserving rollout stability. Extensive evaluations on a benchmark cloth simulation dataset demonstrate that FNOpt outperforms prior learning-based approaches in out-of-distribution settings in both accuracy and robustness. These results position FNO-based meta-optimization as a compelling alternative to previous neural simulators for cloth; thus reducing the need for curated data and improving cross-resolution reliability. Our code is publicly available at https://github.com/Simonhfls/FNOpt.

A potent class of generative models known as Diffusion Probabilistic Models (DPMs) has become prominent. A forward diffusion process adds gradually noise to data, while a model learns to gradually denoise. Sampling from pre-trained DPMs is obtained by solving differential equations (DE) defined by the learnt model, a process which has shown to be prohibitively slow. Numerous efforts on speeding-up this process have consisted on crafting powerful ODE solvers. Despite being quick, such solvers do not usually reach the optimal quality achieved by available slow SDE solvers. Our goal is to propose SDE solvers that reach optimal quality without requiring several hundreds or thousands of NFEs to achieve that goal. We propose Stochastic Explicit Exponential Derivative-free Solvers (SEEDS), improving and generalizing Exponential Integrator approaches to the stochastic case on several frameworks. After carefully analyzing the formulation of exact solutions of diffusion SDEs, we craft SEEDS to analytically compute the linear part of such solutions. Inspired by the Exponential Time-Differencing method, SEEDS use a novel treatment of the stochastic components of solutions, enabling the analytical computation of their variance, and contains high-order terms allowing to reach optimal quality sampling $\sim3$-$5\times$ faster than previous SDE methods. We validate our approach on several image generation benchmarks, showing that SEEDS outperform or are competitive with previous SDE solvers. Contrary to the latter, SEEDS are derivative and training free, and we fully prove strong convergence guarantees for them.