Ordinary Differential Equation
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Related Articles from SNS
Foundation Inference Models for Ordinary Differential Equations
Announce Type: replace Abstract: Ordinary differential equations (ODEs) are central to scientific modelling, but inferring their vector fields from noisy trajectories remains challenging. Current approaches such as symbolic regression, Gaussian process (GP) regression, and Neural ODEs often require complex training pipelines and substantial machine learning expertise, or they depend strongly on system-specific prior knowledge. We propose FIM-ODE, a pretrained Foundation Inference Model that...
Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations
arXiv:2601.09900v4 Announce Type: replace Abstract: This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence.
Hybrid Neural Ordinary Differential Equations for Data-Efficient Polymerization Modeling with Incomplete Kinetics
Announce Type: new Abstract: Accurate prediction of polymerization dynamics is essential for process design, control, and optimization. Yet, purely mechanistic models require labor-intensive parameterization of partially characterized kinetics, while purely data-driven models demand large, diverse datasets that are costly to obtain, particularly in early-design stages. We propose a hybrid Neural Ordinary Differential Equation (NODE) framework for data-efficient modeling of free-radical...
Stochastic Differential Equations (SDEs) in NONMEM for Probing Population Pharmacokinetic Model Misspecification: Diagnostic Utility, Practical Considerations, and Future Directions
Population pharmacokinetic (popPK) models are commonly developed using ordinary differential equations (ODEs) to describe deterministic concentration-time profiles, with unexplained variability typically attributed to interindividual variability or residual error. When model misspecification is present, system-level deviations may be absorbed into these conventional variability terms, making the source and magnitude of model inadequacy difficult to assess quantitatively. Stochastic...
Sparse Discovery of Functional Relationships in Solutions to Systems of Differential Equations
Announce Type: replace-cross Abstract: This work develops a framework to discover relations between the components of the solution to a given initial-value problem for a first-order system of ordinary differential equations. This is done by using sparse identification techniques on the data represented by the numerical solution of the initial-value problem at hand.
Constrained Control of PDE Traffic Flow via Spatial Control Barrier Functions
Announce Type: replace Abstract: In this paper, a constrained control approach to variable speed limit (VSL) control for macroscopic partial differential equations (PDE) traffic models is developed. Control Lyapunov function (CLF) theory for ordinary differential equations (ODE) is extended to account for spatially and temporally varying states and control inputs. The stabilizing CLF is then unified with safety constraints through the introduction of spatially varying control barrier...
Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs
arXiv:2604.03399v2 Announce Type: replace-cross Abstract: Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as...
From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models
arXiv:2606.08956v1 Announce Type: new Abstract: Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the governing differential equation, which is then calibrated with data by solving an inverse problem. In recent years, the field of Scientific Machine...
Stable and Scalable Probabilistic Numerical Solvers for Stiff and High-Dimensional ODEs
arXiv:2606.08203v1 Announce Type: new Abstract: Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this...
A Kinetic Energy Perspective of Flow Matching
arXiv:2602.07928v2 Announce Type: replace Abstract: Flow-based generative models can be viewed through a physics lens: sampling transports a particle from noise to data by integrating a learned velocity field, and each sample corresponds to a trajectory with its own dynamical effort. Motivated by classical mechanics, we introduce Kinetic Path Energy (KPE), an action-like, per-sample diagnostic that measures the accumulated kinetic effort along an ordinary differential equation (ODE)...