This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.
Diffusion probabilistic models generate samples by learning to reverse a noise-injection process that transforms data into noise. Reformulating this reverse process as a deterministic probability flow ordinary differential equation (ODE) enables efficient sampling using high-order solvers, often requiring only $\mathcal{O}(10)$ steps. Since the score function is typically approximated by a neural network, analyzing the interaction between its regularity, approximation error, and numerical integration error is key to understanding the overall sampling accuracy. In this work, we continue our analysis of the convergence properties of the deterministic sampling methods derived from probability flow ODEs [25], focusing on $p$-th order (exponential) Runge-Kutta schemes for any integer $p \geq 1$. Under the assumption that the first and second derivatives of the approximate score function are bounded, we develop $p$-th order (exponential) Runge-Kutta schemes and demonstrate that the total variation distance between the target distribution and the generated data distribution can be bounded above by \begin{align*} O\bigl(d^{\frac{7}{4}}\varepsilon_{\text{score}}^{\frac{1}{2}} +d(dH_{\max})^p\bigr), \end{align*} where $\varepsilon^2_{\text{score}}$ denotes the $L^2$ error in the score function approximation, $d$ is the data dimension and $H_{\max}$ represents the maximum step size used in the solver. We numerically verify the regularity assumption on benchmark datasets, confirming that the first and second derivatives of the approximate score function remain bounded in practice. Our theoretical guarantees hold for general forward processes with arbitrary variance schedules.
We present ODE-GS, a novel method that unifies 3D Gaussian Splatting with latent neural ordinary differential equations (ODEs) to forecast dynamic 3D scenes far beyond the time span seen during training. Existing neural rendering systems - whether NeRF- or 3DGS-based - embed time directly in a deformation network and therefore excel at interpolation but collapse when asked to predict the future, where timestamps are strictly out-of-distribution. ODE-GS eliminates this dependency: after learning a high-fidelity, time-conditioned deformation model for the training window, we freeze it and train a Transformer encoder that summarizes past Gaussian trajectories into a latent state whose continuous evolution is governed by a neural ODE. Numerical integration of this latent flow yields smooth, physically plausible Gaussian trajectories that can be queried at any future instant and rendered in real time. Coupled with a variational objective and a lightweight second-derivative regularizer, ODE-GS attains state-of-the-art extrapolation on D-NeRF and NVFI benchmarks, improving PSNR by up to 10 dB and halving perceptual error (LPIPS) relative to the strongest baselines. Our results demonstrate that continuous-time latent dynamics are a powerful, practical route to photorealistic prediction of complex 3D scenes.
Large Reasoning Models (LRMs) demonstrate remarkable problem-solving capabilities through extended Chain-of-Thought (CoT) reasoning but often produce excessively verbose and redundant reasoning traces. This inefficiency incurs high inference costs and limits practical deployment. While existing fine-tuning methods aim to improve reasoning efficiency, assessing their efficiency gains remains challenging due to inconsistent evaluations. In this work, we introduce the reasoning efficiency frontiers, empirical upper bounds derived from fine-tuning base LRMs across diverse approaches and training configurations. Based on these frontiers, we propose the Reasoning Efficiency Gap (REG), a unified metric quantifying deviations of any fine-tuned LRMs from these frontiers. Systematic evaluation on challenging mathematical benchmarks reveals significant gaps in current methods: they either sacrifice accuracy for short length or still remain inefficient under tight token budgets. To reduce the efficiency gap, we propose REO-RL, a class of Reinforcement Learning algorithms that minimizes REG by targeting a sparse set of token budgets. Leveraging numerical integration over strategically selected budgets, REO-RL approximates the full efficiency objective with low error using a small set of token budgets. Through systematic benchmarking, we demonstrate that our efficiency metric, REG, effectively captures the accuracy-length trade-off, with low-REG methods reducing length while maintaining accuracy. Our approach, REO-RL, consistently reduces REG by >=50 across all evaluated LRMs and matching Qwen3-4B/8B efficiency frontiers under a 16K token budget with minimal accuracy loss. Ablation studies confirm the effectiveness of our exponential token budget strategy. Finally, our findings highlight that fine-tuning LRMs to perfectly align with the efficiency frontiers remains an open challenge.
A novel splitting algorithm is proposed for the numerical simulation of neuromorphic circuits. The algorithm is grounded in the operator-theoretic concept of monotonicity, which bears both physical and algorithmic significance. The splitting exploits this correspondence to translate the circuit architecture into the algorithmic architecture. The paper illustrates the many advantages of the proposed operator-theoretic framework over conventional numerical integration for the simulation of multiscale hierarchical events that characterize neuromorphic behaviors.
Discretizing continuous-time linear systems typically requires numerical integration. This document presents a convenient method for discretizing the dynamics, input, and process noise state-space matrices of a continuous-time linear system using a single matrix exponential.
Molecular dynamics (MD) is a powerful tool for exploring the behavior of atomistic systems, but its reliance on sequential numerical integration limits simulation efficiency. We present MDtrajNet-1, a foundational AI model that directly generates MD trajectories across chemical space, bypassing force calculations and integration. This approach accelerates simulations by up to two orders of magnitude compared to traditional MD, even those enhanced by machine-learning interatomic potentials. MDtrajNet-1 combines equivariant neural networks with a Transformer-based architecture to achieve strong accuracy and transferability in predicting long-time trajectories for both known and unseen systems. Remarkably, the errors of the trajectories generated by MDtrajNet-1 for various molecular systems are close to those of the conventional ab initio MD. The model's flexible design supports diverse application scenarios, including different statistical ensembles, boundary conditions, and interaction types. By overcoming the intrinsic speed barrier of conventional MD, MDtrajNet-1 opens new frontiers in efficient and scalable atomistic simulations.
We present a continuous-time equivalent to the well-known iterative linear-quadratic algorithm including an implementation of a backtracking line-search policy and a novel regularization approach based on the necessary conditions in the Riccati pass of the linear-quadratic regulator. This allows the algorithm to effectively solve trajectory optimization problems with non-convex cost functions, which is demonstrated on the cart-pole swing-up problem. The algorithm compatibility with state-of-the-art suites of numerical integration solvers allows for the use of high-order adaptive-step methods. Their use results in a variable number of time steps both between passes of the algorithm and across iterations, maintaining a balance between the number of function evaluations and the discretization error.
We present closed analytical approximations for the pricing of Asian basket spread options under the Black-Scholes model. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be highly accurate for Asian and spread options in practice. Unlike other approaches, they do not require any numerical integration or root solving.
The paper focuses on the problem of tracking eigenvalue trajectories in large-scale power system models as system parameters vary. A continuation-based formulation is presented for tracing any single eigenvalue of interest, which supports sparse matrix representations and accommodates both explicit and semi-implicit differential-algebraic models. Key implementation aspects, such as numerical integration, matrix updates, derivative approximations, and handling defective eigenvalues, are discussed in detail and practical recommendations are duly provided. The tracking approach is demonstrated through a comprehensive case study on the IEEE 39-bus system, as well as on a realistic dynamic model of the Irish transmission system.
Kernel mean embeddings -- integrals of a kernel with respect to a probability distribution -- are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or for statistical inference based on the maximum mean discrepancy. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.
This paper introduces patch assignment flows for metric data labeling on graphs. Labelings are determined by regularizing initial local labelings through the dynamic interaction of both labels and label assignments across the graph, entirely encoded by a dictionary of competing labeled patches and mediated by patch assignment variables. Maximal consistency of patch assignments is achieved by geometric numerical integration of a Riemannian ascent flow, as critical point of a Lagrangian action functional. Experiments illustrate properties of the approach, including uncertainty quantification of label assignments.
Numerous applications necessitate the computation of numerical solutions to differential equations across a wide range of initial conditions and system parameters, which feeds the demand for efficient yet accurate numerical integration methods.This study proposes a neural network (NN) enhancement of classical numerical integrators. NNs are trained to learn integration errors, which are then used as additive correction terms in numerical schemes. The performance of these enhanced integrators is compared with well-established methods through numerical studies, with a particular emphasis on computational efficiency. Analytical properties are examined in terms of local errors and backward error analysis. Embedded Runge-Kutta schemes are then employed to develop enhanced integrators that mitigate generalization risk, ensuring that the neural network's evaluation in previously unseen regions of the state space does not destabilize the integrator. It is guaranteed that the enhanced integrators perform at least as well as the desired classical Runge-Kutta schemes. The effectiveness of the proposed approaches is demonstrated through extensive numerical studies using a realistic model of a wind turbine, with parameters derived from the established simulation framework OpenFast.
Molecular dynamics (MD) simulations play a crucial role in scientific research. Yet their computational cost often limits the timescales and system sizes that can be explored. Most data-driven efforts have been focused on reducing the computational cost of accurate interatomic forces required for solving the equations of motion. Despite their success, however, these machine learning interatomic potentials (MLIPs) are still bound to small time-steps. In this work, we introduce TrajCast, a transferable and data-efficient framework based on autoregressive equivariant message passing networks that directly updates atomic positions and velocities lifting the constraints imposed by traditional numerical integration. We benchmark our framework across various systems, including a small molecule, crystalline material, and bulk liquid, demonstrating excellent agreement with reference MD simulations for structural, dynamical, and energetic properties. Depending on the system, TrajCast allows for forecast intervals up to $30\times$ larger than traditional MD time-steps, generating over 15 ns of trajectory data per day for a solid with more than 4,000 atoms. By enabling efficient large-scale simulations over extended timescales, TrajCast can accelerate materials discovery and explore physical phenomena beyond the reach of traditional simulations and experiments. An open-source implementation of TrajCast is accessible under https://github.com/IBM/trajcast.
Engineered systems naturally experience large disturbances that can disrupt desired operation because the system may fail to recover to a stable equilibrium point. It is valuable to determine the mechanism of instability when the system is subject to a particular finite-time disturbance, because this information can be used to improve vulnerability detection, and to design controllers to reduce vulnerability. Often there exists a particular unstable equilibrium point on the region of attraction boundary of the stable equilibrium point such that the unstable eigenvector of the Jacobian at this unstable equilibrium point represents the mode of instability for the disturbance. Unfortunately, it is challenging to find this mode of instability, especially in high dimensional systems, because it is computationally intractable to obtain this particular unstable equilibrium point. This paper develops a novel algorithm for numerically computing the mode of instability for parameterized nonlinear systems without identifying the particular unstable equilibrium point, resulting in a computationally efficient method. The key idea is to first consider the setting where the system recovers, and to average the Jacobian along the system trajectory from the post-disturbance state up until the Jacobian becomes stable. As the system approaches inability to recover, the averaged Jacobians converge to the Jacobian at the particular unstable equilibrium point, and can be used to extract the unstable eigenvector representing the mode of instability. Convergence guarantees are provided for computing the mode of instability, both for the theoretical setting in continuous time, and for the proposed algorithm which relies on numerical integration. Numerical examples illustrate the successful application of the method to identify the mechanism of instability in power systems subject to temporary short circuits.
Dynamic simulation plays a crucial role in power system transient stability analysis, but traditional numerical integration-based methods are time-consuming due to the small time step sizes. Other semi-analytical solution methods, such as the Differential Transformation method, often struggle to select proper orders and steps, leading to slow performance and numerical instability. To address these challenges, this paper proposes a novel adaptive dynamic simulation approach for power system transient stability analysis. The approach adds feedback control and optimization to selecting the step and order, utilizing the Differential Transformation method and a proportional-integral control strategy to control truncation errors. Order selection is formulated as an optimization problem resulting in a variable-step-optimal-order method that achieves significantly larger time step sizes without violating numerical stability. It is applied to three systems: the IEEE 9-bus, 3-generator system, IEEE 39-bus, 10-generator system, and a Polish 2383-bus, 327-generator system, promising computational efficiency and numerical robustness for large-scale power system is demonstrated in comprehensive case studies.
Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.
We provide a general framework for learning diffusion bridges that transport prior to target distributions. It includes existing diffusion models for generative modeling, but also underdamped versions with degenerate diffusion matrices, where the noise only acts in certain dimensions. Extending previous findings, our framework allows to rigorously show that score matching in the underdamped case is indeed equivalent to maximizing a lower bound on the likelihood. Motivated by superior convergence properties and compatibility with sophisticated numerical integration schemes of underdamped stochastic processes, we propose \emph{underdamped diffusion bridges}, where a general density evolution is learned rather than prescribed by a fixed noising process. We apply our method to the challenging task of sampling from unnormalized densities without access to samples from the target distribution. Across a diverse range of sampling problems, our approach demonstrates state-of-the-art performance, notably outperforming alternative methods, while requiring significantly fewer discretization steps and no hyperparameter tuning.
The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands to capture the uncertainty arising from the sparse function evaluations. Quantum computers have emerged as potential replacements for classical computers, offering exponential reductions in the computational complexity of machine learning tasks. In this paper, we combine Gaussian process quadratures and quantum computing by proposing a quantum low-rank Gaussian process quadrature method based on a Hilbert space approximation of the Gaussian process kernel and enhancing the quadrature using a quantum circuit. The method combines the quantum phase estimation algorithm with the quantum principal component analysis technique to extract information up to a desired rank. Then, Hadamard and SWAP tests are implemented to find the expected value and variance that determines the quadrature. We use numerical simulations of a quantum computer to demonstrate the effectiveness of the method. Furthermore, we provide a theoretical complexity analysis that shows a polynomial advantage over classical Gaussian process quadrature methods. The code is available at https://github.com/cagalvisf/Quantum_HSGPQ.
Parameter estimation and trajectory reconstruction for data-driven dynamical systems governed by ordinary differential equations (ODEs) are essential tasks in fields such as biology, engineering, and physics. These inverse problems -- estimating ODE parameters from observational data -- are particularly challenging when the data are noisy, sparse, and the dynamics are nonlinear. We propose the Eigen-Fourier Physics-Informed Gaussian Process (EFiGP), an algorithm that integrates Fourier transformation and eigen-decomposition into a physics-informed Gaussian Process framework. This approach eliminates the need for numerical integration, significantly enhancing computational efficiency and accuracy. Built on a principled Bayesian framework, EFiGP incorporates the ODE system through probabilistic conditioning, enforcing governing equations in the Fourier domain while truncating high-frequency terms to achieve denoising and computational savings. The use of eigen-decomposition further simplifies Gaussian Process covariance operations, enabling efficient recovery of trajectories and parameters even in dense-grid settings. We validate the practical effectiveness of EFiGP on three benchmark examples, demonstrating its potential for reliable and interpretable modeling of complex dynamical systems while addressing key challenges in trajectory recovery and computational cost.