numerical sde simulation

Adaptive stochastic differential equation (SDE) solvers lack methods for exact sampling of higher-order Brownian time integrals—such as ∫Wᵣ dr—essential for achieving high-order convergence. Method: We propose the extended Virtual Brownian Tree (VBT), the first framework enabling joint exact sampling of both Brownian paths and their time integrals. Implemented in JAX, it employs a single-seed pseudorandom number generator to realize a memory-constant, error-controlled, reproducible binary-search VBT, with theoretical guarantees of strict distributional consistency at ε-separated query points. The implementation is integrated into the Diffrax library. Results: Experiments demonstrate that our method more than doubles the observed convergence order of adaptive high-order SDE solvers on a highly volatile CIR model. In MCMC tasks, a third-order kinetic Langevin solver using our method outperforms NUTS, reducing function evaluations by 90%.

Approximating functional solutions of stochastic differential equations (SDEs) remains challenging due to the “curse of dimensionality” inherent in traditional polynomial chaos expansions (PCE). Method: We propose SDEONet—the first framework integrating Wiener chaos expansion (WCE) with deep operator networks (DeepONets)—to enable end-to-end functional approximation of the SDE solution operator. SDEONet employs sparse learning to automatically identify optimal low-dimensional truncations of WCE, circumventing ad hoc truncation and preserving statistical fidelity. Contributions/Results: (i) First systematic incorporation of WCE into SDE solvers; (ii) rigorous proof of convergence and subexponential computational complexity; (iii) scalability to high-dimensional, nonlinear SDEs beyond sampling-based methods. Numerical experiments demonstrate substantial accuracy and efficiency gains over Monte Carlo—especially for 1D to high-dimensional SDEs—with exponential speedup in computation and built-in uncertainty quantification.

Diffusion probabilistic models (DPMs) require solving computationally expensive diffusion stochastic differential equations (SDEs) or ordinary differential equations (ODEs) during sampling; existing acceleration methods predominantly target deterministic ODE solvers, struggling to balance generation fidelity and diversity. This work proposes the first stochastic Adams linear multistep solver tailored for diffusion SDEs, jointly optimizing fidelity and diversity with minimal function evaluations (20–50 NFEs) via variance-controlled SDE modeling, a novel stochastic integrator design, and NFE-aware scheduling. Its core innovation lies in the first integration of the stochastic Adams method with the linear multistep framework for SDE solving. On benchmarks including CIFAR-10 and CelebA-HQ, our method achieves state-of-the-art FID scores in only 4–8 sampling steps—substantially outperforming both existing deterministic and stochastic samplers.

Diffusion model sampling faces a fundamental trade-off between speed and sample quality: ODE solvers are efficient but suboptimal in performance, whereas SDE solvers achieve superior quality at high computational cost. To address this, we propose the Extended Reverse-time Stochastic Differential Equation (ER-SDE) framework—a unified theoretical model that reveals the intrinsic cause of performance disparity between ODE and SDE samplers as one-step prediction error. We derive the first exact and approximate analytical solutions for VP and VE SDEs within this framework and prove the equivalence of classical ODE and SDE solvers under ER-SDE. Leveraging semilinear SDE theory and reverse-time modeling, we design efficient numerical solvers—ER-SDE-Solvers—that inherit both deterministic efficiency and stochastic expressiveness. Evaluated on ImageNet 128×128, our method achieves a state-of-the-art FID of 8.33 in only 20 function evaluations, bridging the gap between speed and generation quality.

A theoretical-practical gap persists in score-based diffusion models. Method: We propose a unified, reproducible SDE-based modeling framework that systematically integrates score matching, SDE/ODE solvers, denoising score estimation, and consistency modeling; notably, we introduce reinforcement learning into diffusion sampling for inference-path optimization. Contributions: (1) We establish theoretical consistency between sampling and score estimation under the SDE formulation; (2) we provide concise proofs of key theorems alongside practical algorithmic implementation guidelines; (3) we release modular, open-source code enabling rapid validation and extension to novel architectures. This work bridges the efficiency of score matching with scalable, RL-enhanced inference, delivering a foundational toolkit that balances theoretical rigor and engineering practicality for the design, analysis, and application of diffusion models.

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 computationally expensive inverse problem of parameter estimation for stochastic differential equations (SDEs) by proposing an efficient solution framework that, for the first time, integrates Wiener chaos expansion (WCE) with stochastic gradient descent (SGD). By projecting the stochastic solution onto a deterministic system of propagators via an orthogonal Hermite polynomial basis, the method constructs a regularized discrepancy functional amenable to SGD optimization. This transformation effectively converts the original stochastic inverse problem into a deterministic optimization task, substantially reducing computational complexity and data requirements. Numerical experiments on several nonlinear SDE models—including a biological individual growth model—demonstrate that the approach accurately and robustly recovers parameters even from sparse and noisy observational data, exhibiting strong scalability and practical promise.

Multidimensional stochastic differential equations (SDEs) generally lack closed-form solutions, and existing numerical methods are often constrained in strong convergence order and computational efficiency, particularly when handling multiple stochastic integrals where accuracy and complexity are difficult to balance. This work proposes an improved Milstein scheme that incorporates two novel algorithms for efficiently computing multiple stochastic integrals and establishes a theoretical framework enabling verifiable strong and weak convergence orders. The method accurately assesses convergence performance even in the absence of analytical solutions, significantly enhancing both accuracy and efficiency for high-dimensional SDEs. Numerical experiments and applications to financial models demonstrate that the proposed approach outperforms current techniques in convergence rate and computational cost, offering a highly accurate and scalable numerical tool for simulating high-dimensional stochastic systems.

Bayesian parameter inference for ordinary differential equation (ODE) models from observational data often neglects discretization error introduced by numerical solvers, leading to overconfident and biased posterior estimates. Method: We propose a joint uncertainty quantification framework that models discretization error as a time-evolving stochastic process. Leveraging an asymptotically justified Markov prior, we explicitly encode its variance structure; integrating a state-space model with randomized numerical solvers enables simultaneous Bayesian inference of both ODE parameters and discretization error. Contribution/Results: Experiments demonstrate that our approach significantly broadens the support of the parameter posterior distribution while accurately disentangling and quantifying uncertainties attributable to parameters versus discretization. This enhances robustness and interpretability of ODE models under sparse and noisy observations.

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.