Ordinary Differential Equations

Also known as ODE

Follow this topic Ask about this topic

Latest papers

33 papers3 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

Sep 29, 2026

Identifying ODEs from Unstructured Data with Causal Representation Learning

We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and equations. While Causal Representation Learning (CRL) methods provide guarantees on identifying variables from high-dimensional observations up to component-wise diffeomorphisms, we show that in general these variables cannot be used directly as input to equation discovery methods, which typically assume that the variables will lead to sparse equations. So we introduce SParse Equivalent Equation Discovery AutoEncoder (SPEED-AE), a framework that combines a pretrained CRL method with a component-wise autoencoder that learns transformations of variables that are amenable to sparse ODE discovery. We show that for polynomial ODEs, this additional step allows us to restrict the identifiability of each variable from polynomial to monomial diffeomorphisms. Experiments on Lotka-Volterra, Lorenz, and a two-pendulum system show that SPEED-AE improves on the disentanglement of the CRL methods and that it recovers ODEs that are closest to the ground truth, while achieving state-of-the-art forecasting performance.

Alessandro Trenta, Riccardo Massidda, Davide Bacciu +1

Department of Computer Science University of Pisa · Saarland Informatics Campus Saarland University & University of Amsterdam

Causal Representation Learning Representation Identifiability Sparse Identification of Nonlinear Dynamics ODE Discovery Ordinary Differential Equations

Sep 22, 2026

TinyUDE: Solver-Free Universal Differential Equations on Microcontrollers via Lie-Taylor Jet Matching

Training Universal Differential Equations (UDEs) traditionally relies on backpropagating through numerical ODE solvers, creating memory footprints far exceeding the capabilities of edge microcontrollers. We present Lie-Taylor jet matching, a solver-free training framework that fits a hybrid vector field directly to the first and second time-derivatives of observed system states. These derivatives, the truncated Lie-Taylor jet, are estimated online via Savitzky-Golay filtering, yielding fully analytic gradients without automatic differentiation software. We evaluate whether eliminating the solver compromises accuracy against a conventional baseline (fixed-step RK4 integration, multiple shooting, exact discrete adjoints, Adam) sharing identical dynamics, noise models, network architectures, and metrics. While naive derivative matching degrades under sensor noise, our noise-adaptive mechanisms close and reverse this gap: full-rate phase-shifted sampling, a reservoir buffer, cosine-annealed optimization with weight averaging, on-device noise estimation, and polynomial-misfit quality gating. On a damped pendulum and chaotic double pendulum, our method matches or exceeds baseline accuracy at matched data windows and recovers unmodeled damping coefficients. Across noise levels from 0% to 5%, it attains a geometric-mean relative field error of 0.65x that of the baseline within 108 kB of static memory, compared with megabytes of solver tape. On an ESP32 microcontroller, the on-device run reaches a field error of 0.0020 and recovers the damping coefficient to c = 0.400 (true 0.400) within 61.3 kB of static memory and 7.24 ms per update (18.1% duty cycle at 25 Hz), confirming real-time on-device training is feasible without a numerical solver.

Pranavanath Balamurali, Hrishi Kamireddy

Department of Electrical and Computer Engineering The University of Texas at Austin · Department of Computer Science The University of Texas at Austin

On-Device Training System Identification Ordinary Differential Equations

Sep 21, 2026

Enhancing Transformer Representations of Symbolic ODE Expressions

Existing approaches to solving differential equations, such as symbolic regression, physics informed neural networks, and neural operators, typically focus on numerical approximations or blind symbolic search via fitting to numerical data. Less attention has been paid to learning structured representations of mathematical expressions that preserve commutative properties and could support mathematical reasoning in symbolic forms. Transformer models have shown strong capabilities in solving symbolic differential equations. However, standard positional embeddings in transformers are designed for sequence data. Symbolic differential equations are naturally represented by expression trees, so these positional embeddings may not efficiently capture their hierarchical structures. We investigate existing tree positional embeddings in symbolic ordinary differential equation (ODE) tasks. We systematically study their effectiveness under different settings. Our results show that tree positional embeddings aid learning in early epochs and continue to improve performance throughout, ultimately yielding consistent advantages across various data sizes and tasks. Based on learned structural representations, we apply contrastive learning to support the commutative property in mathematics. Ablation studies provide insight into how these methods interact in modelling symbolic mathematical structures.

Xiyue Fan, Adam Prugel-Bennett, Stuart E. Middleton

University of Southampton

Transformer Positional Encoding Ordinary Differential Equations

Sep 3, 2026

Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis

In this paper, the problem of data-driven discovery of nonlinear ordinary differential equations (ODEs) is recast, and a new interpretable machine learning (ML) method is proposed. The proposed method aims to learn the unknown vector field of nonlinear dynamics without prior knowledge of the system's physics from only one single state trajectory's data. The proposed method has two fundamental differences with existing methods: 1) the formulation presented in this method is derived based on Functional Analysis and Operator Theory, and 2) the cost function is constructed in the function space as a distance between two functions as an integral, instead of the discrete-sum of errors used in existing ML approaches. An incremental learning algorithm is proposed to learn the unknown vector field to handle new training samples in an online manner. The proposed method can discover the unknown vector field from both forced and unforced autonomous and non-autonomous (or time-varying) dynamical systems. The proposed method is able to simultaneously discover unknown external forces as a function of time and unknown underlying dynamics. Finally, numerical examples are given to demonstrate the advantages of the proposed method.

Seyyed Shaho Alaviani, Yongzhi Qu, Gregory W. Vogl

Department of Mechanical Engineering University of Minnesota-Twin Cities Minneapolis, MN 55455, USA · Department of Mechanical Engineering University of Utah Salt Lake City, UT 84112, USA · National Institute of Standards and Technology Gaithersburg, MD 20899, USA

ODE Discovery Nonlinear System Identification Ordinary Differential Equations Online System Identification

Sep 3, 2026

Time Without Timesteps: Simulating Coupled Dynamical Systems via Self-Consistency

Numerical simulation of dynamical systems is usually organized as a causal march through time: each state is computed from the previous one. We explore a different formulation for coupled systems. For each subsystem type we train a neural surrogate mapping a full driving trajectory and initial condition directly to a full output trajectory; following classical waveform relaxation, coupled systems are assembled by enforcing self-consistency among these trajectories: simulation becomes a fixed-point problem over complete trajectories rather than a stepwise rollout. On coupled van der Pol oscillators and Hodgkin-Huxley neuron networks, sequential depth becomes the number of solver iterations: 4-10 Newton iterations where the reference integrator takes 1500 steps. The gradient likewise loses its time recursion: it becomes a linear system solved by GMRES at memory independent of solver depth. A single scalar measured from the learned operator, the spectral radius of its Jacobian, predicts in advance where the coupled solve will converge; past that boundary, unrolled backpropagation diverges and a Neumann adjoint fails, while the implicit gradient remains correct to 0.04%. We report where the approach succeeds and where surrogate error degrades it.

Liyu Zerihun, Mark Shinyoung Lee

Dynamical Systems Neural Operators Ordinary Differential Equations

Aug 22, 2026

Symbolic Neural ODEs: Learning interpretable models from time-series data

We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.

Nibodh Boddupalli, Jeff Moehlis

Dept. of Mechanical Engineering, University of California, Santa Barbara, CA 93106

Nonlinear System Identification System Identification Neural ODEs Sparse Identification of Nonlinear Dynamics Ordinary Differential Equations Chaotic Time-Series Prediction

Aug 14, 2026

GradRepair-ODE: Certified Gradient Repair for Neural ODE Training

Neural ordinary differential equations use numerical solvers inside the training loop. The solver determines the forward trajectory and also affects the gradient passed to the optimizer. That coupling creates a reliability problem for scientific machine learning and continuous-time generative modeling, including diffusion probability-flow ordinary differential equations and flow-matching models. Under loose step sizes, stiff dynamics, chaotic sensitivity, or event discontinuities, a differentiable ODE pipeline can return a finite gradient whose direction is numerically suspect. We introduce GradRepair-ODE, a reliability framework for checking, repairing, and rejecting ODE gradients at the optimizer step. The method computes several gradient candidates, compares them with directional finite-difference checks and solver diagnostics, diagnoses likely numerical failure modes, repairs selected gradients through path switching or stricter recomputation, and rejects steps whose descent direction cannot be certified. In six synthetic ODE systems, GradRepair-ODE leaves low-risk systems unchanged, repairs Robertson and Lorenz gradients to cosine similarity 1.000 against a strict reference, reduces unsafe accepted steps from 37 to 0, and rejects an event-discontinuous case instead of applying an uncertified update. The paper argues for a simple change in the training contract: an ODE gradient should reach the optimizer with numerical evidence attached.

Ziqian Bi, Xin Liang Chia

Purdue University, West Lafayette, IN, USA · Rice University, Houston, TX, USA

Neural ODEs Ordinary Differential Equations Neural Network Optimization

Aug 13, 2026

Sparse Orthogonal Regression Technique: A Spectral Framework for Equation Discovery, Approximation, and Integration

We develop the Sparse Orthogonal Regression Technique (SORT), a sparse spectral framework for learning orthonormal-basis expansions from noisy and irregularly sampled data. SORT estimates expansion coefficients directly from observations using L1-regularized regression, avoiding explicit quadrature or analytic inner-product evaluation. The central application is data-driven discovery of ordinary differential equations: vector fields are represented in chosen orthogonal bases and learned as sparse coefficient expansions. This provides a complementary route to symbolic regression, grammar-based discovery, and SINDy-style sparse identification by first recovering a compact spectral representation, which can later guide searches for simpler analytic forms. Across the dynamical-system experiments, SORT matches or improves upon library-based sparse-regression baselines when the basis is well adapted to the problem, and shows more stable degradation under sparse sampling, noisy derivative estimates, and representation mismatch. Specific examples illustrate why this representation is useful: if a finite library misses the problem-specific nonlinearity, the resulting model can fail. SORT is not immune to mismatch, but it shifts the problem away from brittle selection among generic terms to basis design adapted to the problem domain. The experiments also show that dominant low-order coefficients persist as model order increases, supporting order-consistent model growth. Beyond equation discovery, the same learned expansion supports nonlinear approximation and estimation of complex, high-dimensional integrals by coefficient readout. Overall, SORT provides a reusable intermediate representation for system identification, approximation, and integration, while making basis design an explicit part of the scientific modeling problem.

Sabin Roman, Ljupco Todorovski, Saso Dzeroski

Department of Knowledge Technologies, Jožef Stefan Institute · Faculty of Mathematics and Physics, University of Ljubljana

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Orthogonal Representation Learning ODE Discovery Ordinary Differential Equations

Aug 10, 2026

Marrying Optimal Transport and ODEs for Unified Continuous-Time 4D Reconstruction and Tracking

Existing unified 4D reconstruction and point tracking approaches typically rely on heuristic interpolations or just predict at integer timestamps, lacking kinematic coherence and failing to model dynamics at any arbitrary timestamp. In this paper, we propose Uni4R, a framework that unifies these tasks by learning continuous velocity fields through the synergy of Optimal Transport (OT) and Ordinary Differential Equation (ODE). Importantly, this continuous velocity field acts as a kinematic prior that mutually benefits both 4D reconstruction and point tracking. Specifically, we propose the Flow Matching Guided Decoder (FMGD). A global velocity branch first extracts anchor features that capture the global dynamic state of the sequence. Then, FMGD leverages Flow Matching (FM) theory to formulate a probability path defined by OT on the anchor feature manifold, instantiating it as FM-guided velocity features for velocity prediction. This establishes a robust kinematic inductive bias. Meanwhile, a point reconstruction branch provides geometric features. The local velocity prediction module then joint above features and time embeddings, to decode velocities at arbitrary timestamps. To overcome the absence of high-quality ground-truth velocities in fractional frames, we propose an integral-consistency training strategy. This strategy uses an ODE solver to integrate velocities to recover target pointmaps, enabling the model to be supervised end-to-end directly from integer timestamps. Experimental results demonstrate that Uni4R achieves SOTA performance in both 4D reconstruction and point tracking, and achieves SOTA in our new kinematics-aware benchmark at continuous time.

Liying Yang, Hao Mo, Jialun Liu +7

SCSE, Macau University of Science and Technology · The University of Queensland · Great Bay University +3

Point Tracking 4D Reconstruction Ordinary Differential Equations

Aug 1, 2026

HyperODE: Zero-Shot Surrogate for Simulation and Inference of Dynamical Systems

Understanding and controlling complex dynamical systems often requires executing thousands of numerical simulations across vast parametric landscapes, which is time-consuming. Machine learning surrogates significantly accelerate simulation by predicting state trajectories across different initializations and parameter values. However, surrogate models are specialized to one simulation model. Modifying the underlying differential equations - e.g., adding a physiological state or altering an epidemiological contact network - renders trained models obsolete and forces computationally expensive retraining from scratch. We introduce HyperODE, a surrogate capable of operating across an entire class of approximately mass-conserving compartmental models without retraining. By mapping the structure of ordinary differential equations (ODEs) into directed hypergraphs, HyperODE decouples the functional form of system interactions from the neural network architecture. HyperODE takes a compartmental model in the form of an ODE with an arbitrary parameter distribution defined through quantiles and transforms it into a hypergraph. It outputs the distribution of the trajectories for all the states in the original ODE in the form of quantiles. We then use this surrogate to build an encoder that takes a noisy trajectory and outputs a distribution over the parameters of the original ODE, thus calibrating the model in a single pass. On families and system sizes never seen in training, HyperODE produces calibrated quantile bands in a single forward pass, with weighted-interval score and coverage on par with specialized surrogates for each structure. For inverse inference, HyperODE produces calibration from noisy state trajectories in a few milliseconds with a single shared encoder, competitive with existing methods. HyperODE extends zero-shot to ODEs that break mass conservation and to external forcing.

Ajitesh Srivastava

Northeastern University

Neural Surrogate Modeling Zero-Shot Learning Simulation-Based Inference Surrogate Modeling Ordinary Differential Equations

Jul 31, 2026

Dynamics-aware identification of governing equations from sparse and noisy data

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

Pongpisit Thanasutives, Yoshinobu Kawahara

Center for Advanced Intelligence Project (AIP), RIKEN, Osaka, Japan · 2Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan

Koopman Operator Learning Sparse Identification of Nonlinear Dynamics PDE Discovery ODE Discovery Dynamic Mode Decomposition Ordinary Differential Equations

Jul 20, 2026

The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery

How the dynamical regime of the observed system affects equation discovery has mainly been investigated through comparisons across systems. However, such comparisons vary both the equations and the dynamics, confounding the effect of the regime with the difficulty of recovering the equations symbolically. We separate the two by varying the forcing of Lorenz-84, moving its fully observed post-transient trajectories through fixed-point, periodic, and chaotic regimes while preserving the equations' functional form. Within each regime, we separately vary the amount of data, the noise, and prior knowledge of which terms the equations contain. We then measure how well two complementary approaches, sparse regression over a fixed library of candidate functions (SINDy) and an evolutionary search over symbolic expression trees (PySR), recover the true equations' terms and coefficients. We find that recovery depends on whether the sampled states distinguish combinations of candidate functions: equations remain poorly recovered from fixed-point data even when the candidate set contains only the true terms. We link the effect of the dynamics on both algorithms to one object: the moment matrix of the candidate functions under the invariant measure of the regime. Small eigenvalues mark weakly distinguishable combinations of candidate functions: we show that more data, less noise, and more prior knowledge can mitigate the resulting recovery difficulties, while a zero eigenvalue makes distinct equations indistinguishable on the visited states. Hence, a more precise prior needs less informative data, with consequences for data collection, method design, and evaluation.

Matteo Gallo, Fabio Anselmi, Paolo Lazzari

University of Trieste, Trieste, Italy · National Institute of Oceanography and Applied Geophysics - OGS, Trieste, Italy · NBFC, National Biodiversity Future Center, Palermo, Italy

Scientific Discovery Parameter Identifiability Sparse Identification of Nonlinear Dynamics Dynamical Systems Symbolic Regression ODE Discovery+1

Jul 16, 2026

RTS Smoother-Guided Learning of Physics-Based Neural Differential Models

Ordinary differential equations (ODEs) are widely used to model dynamical systems in physics, biology, neuroscience, and physiology, but in many applications some equations of the dynamics are unknown and only a subset of the state variables are measured. We propose a hybrid neural--physics framework in which the known components of the ODE are kept explicit and the missing components are represented by a neural network. The proposed method consists of two stages where we alternate between state and parameter estimation and iterate until a predetermined criterion is met. Specifically, in the first step, we treat the model parameters as being known and we infer the latent states from the available measurements using a Rauch--Tung--Striebel (RTS) smoother. In the second stage, we treat the smoothed trajectories as being known and use them to estimate the neural networks' parameters through backpropagation. We evaluate the method on benchmark systems spanning linear, nonlinear, and stiff dynamics under partial state observation. Across these settings, the proposed method learns missing ODE components from incomplete measurements while exploiting and retaining interpretable mechanistic structure and improving latent-state reconstruction and long-horizon prediction.

Ahmet Demirkaya, Georgios Stratis, Tales Imbiriba +2

Department of Electrical and Computer Engineering Northeastern University · Department of Computer Science University of Massachusetts Boston · Emory University

Nonlinear System Identification Parameter Estimation Latent Dynamics Modeling Dynamical Systems Neural ODEs Physics-Informed ML+1

Jul 15, 2026

Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System

Automatic scientific discovery has long been a goal of computational scholars - a machine that can discover nature's secrets on its own, moving computational systems beyond data-fitting tools toward the generation and refinement of mechanistic models of the universe. Recent advances in symbolic regression (SR) and large-language-model (LLM)-based agents suggest that such systems can recover equations from data, incorporate domain priors, and automate parts of the research workflow. However, most existing approaches either focus on narrow equation-discovery benchmarks or broad end-to-end automation pipelines, while biological systems remain comparatively underexplored. Here, we introduce the MEDA system, an LLM- and SR-powered agentic framework for discovering ordinary-differential-equation (ODE) models of biological and biologically inspired dynamical systems. MEDA retrieves background knowledge, defines admissible variables, generates mechanistic constraints, proposes candidate ODEs, and fits and evaluates them. We evaluate it across canonical model retrieval, reasoning-based extrapolation to unseen variants, and open-ended discovery, with and without experimental data. Across these settings, MEDA recovered the correct state variables, achieved strong structural recovery in retrieval and extrapolation tasks, and produced biologically plausible discovery-oriented models. Ablation and robustness analyses show that knowledge-guided formalization and mechanistic constraints are load-bearing components, whereas numerical fitting alone can preserve trajectory-compatible but biologically incorrect equations.

David Krongauz, Arad Zulti, Eran Segal +1

Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel · Department of Molecular Cell Biology, Weizmann Institute of Science, Rehovot, Israel · Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates +2

Dynamical Systems AI Agents for Scientific Discovery Autonomous Scientific Discovery ODE Discovery Symbolic Regression Ordinary Differential Equations

Jun 25, 2026

Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs

Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem. We establish identifiability bounds for governing ODEs across a wide class of structure equations--ranging from linear ODEs to nonlinear classes with Lipschitz (Hölder)-continuous vector fields--characterizing precisely when two distinct equations can be distinguished from solution data. Using this metric, we derive metric entropy estimates for the relevant ODE classes and analyze sample complexity bounds, quantifying how many solution observations are needed to reliably recover the governing equation.

Yang Pan, Helmut Bölcskei

Parameter Identifiability Sample Complexity Nonlinear System Identification Statistical Learning Theory ODE Discovery Ordinary Differential Equations

Jun 25, 2026

ResilPhase: Plug-and-Play Phase Mapping and Noise-Resilient Macro-Trajectory Extrapolation for Diffusion Acceleration

The adoption of powerful diffusion models is hindered by their significant inference latency. Recent ``cache-then-forecast'' schemes alleviate this issue by accelerating DiTs using derivative-based polynomials, but they suffer from severe quality degradation at high acceleration ratios. Our analysis reveals its root cause: the discrete extrapolation performed on representations that are misaligned with the continuous diffusion trajectory and are numerically unstable. Thus, accelerated DiTs suffer from accumulated spatial errors, noisy derivative amplification, and high-order instability. We therefore reformulate accelerated inference as stable macro-trajectory extrapolation in ordinary differential equation (ODE) space. Instead of predicting intermediate features, we align forecasting with the model's Global Drift (GD), i.e., the end-to-end state evolution, thereby eliminating feature inconsistency and memory overhead. However, even this smooth macro-trajectory remains vulnerable to the derivative fallacy: its higher-order temporal derivatives are intrinsically noisy. Thus, we introduce a derivative-free barycentric Lagrange extrapolator to effectively bypass derivative instability and approximation error. We further propose a bounded Phase Mapping that regularizes the extrapolation domain, suppressing oscillatory error growth. These elements collectively constitute ResilPhase, a noise-resilient acceleration framework. Experiments on FLUX.1-dev and HunyuanVideo demonstrate state-of-the-art fidelity under aggressive acceleration ratios.

Qicheng Zhao, Yu Li, Qi Sun +1

Zhejiang University, Hangzhou, China

Diffusion Transformer Diffusion Sampling Ordinary Differential Equations Diffusion Model Inference Acceleration

Jun 11, 2026

Neural Architectures as Functional Priors in Physics-Informed Control Problems

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

Sonia Rubio Herranz, Fernando Carlos López Hernández, Antonio López Montes

Department of Statistics and Operations Research (EIO), Universidad Complutense de Madrid, 28040 Madrid, Spain · Department of Software Engineering and Artificial Intelligence, Universidad Complutense de Madrid, 28040 Madrid, Spain · Department of Analysis and Applied Mathematics, Universidad Complutense de Madrid, 28040 Madrid, Spain

Dynamical Systems Ordinary Differential Equations Physics-Informed ML Optimal Control

Jun 10, 2026

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds

Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous a posteriori upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two-sided error enclosures. In this paper, we derive computable a posteriori lower bounds for PINN errors in ordinary differential equations on suitable certified state-space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural-network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time-invariant and time-varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed-residual finite-probe certificate based on coordinate unit vectors. We also formulate a certificate-informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post-training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.

Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov

Physikalisch-Technische Bundesanstalt (PTB), Germany · Technical University of Berlin, Germany · Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany +1

Dynamical Systems Neural Network Verification Ordinary Differential Equations Physics-Informed ML

May 26, 2026

PIDM-DP: Physics-Informed Diffusion with Dormand-Prince Integration for Chaotic System Identification and State Reconstruction across Multiple Dynamical Regimes

Reconstructing continuous state trajectories of chaotic dynamical systems from sparse, noisy observations remains a fundamental open problem in nonlinear science. We introduce the Physics-Informed Diffusion Model with Dormand-Prince Integration (PIDM-DP), which embeds a fully differentiable 5th-order Dormand-Prince (DP-RK45) ODE integrator directly into the reverse sampling loop of a Denoising Diffusion Probabilistic Model (DDPM). At each denoising step, physics residuals are back-propagated via automatic differentiation, constraining every generated trajectory to satisfy the system's governing equations to 5th-order accuracy. A linear-scheduled guidance mechanism that ramps the physics weight from zero at high noise levels to its full value near the clean-data limit prevents the gradient explosions that cause naive physics-informed approaches to fail on stiff systems with Jacobian eigenvalues of order

O(10^3)

. Evaluated across five benchmark systems of increasing complexity 3D Lorenz, 3D Rössler, 5D Hyperchaotic, 20D Lorenz-96, and the stiff 3D Rabinovich-Fabrikant at 10% observation density with additive Gaussian noise (

σ=0.05

), PIDM-DP achieves reconstruction RMSE improvements of up to

15.4\times

over an unconstrained diffusion baseline and decisively outperforms the Ensemble Kalman Filter on stiff systems where ensemble covariance collapses. On the Rabinovich-Fabrikant out-of-distribution benchmark, PIDM-DP attains RMSE

0.1097 \pm 0.0269

versus

0.9443 \pm 0.5288

(unconstrained diffusion,

8.6\times

worse) and

0.3561 \pm 0.3040

(EnKF,

3.2\times

worse), with

p<0.001

in paired Wilcoxon tests (

N = 30

). Topological validation via the Rosenstein Lyapunov estimator confirms that PIDM-DP preserves the chaotic invariant measure.

Shailendra Dabral

Indian Institute of Technology Indore

Physics-Informed Diffusion Models Nonlinear System Identification Chaotic Dynamical Systems Physics-Informed ML Diffusion Models Ordinary Differential Equations

May 22, 2026

Learning dynamical systems with biochemically informed neural ordinary differential equations

Ordinary differential equation models of biochemical reactions are often formulated as stoichiometric systems in which the dynamics arise from a collection of interacting processes. A central challenge is that the functional form of each process is rarely known a priori and may be difficult to infer from data. We propose biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE framework that retains the stoichiometric structure of mechanistic models while representing individual processes by neural networks. In BINODEs, the outputs of neural network processes (NNPs) are mapped to state derivatives through a linear layer analogous to a stoichiometric matrix. This architecture allows biological side information, such as process-specific inputs, sign constraints, and monotonicity assumptions, to be built directly into the model. We characterize the approximation properties of NNPs for several standard biochemical rate laws and show that the proposed framework recovers both trajectories and process-level structure in Monod, Lotka--Volterra, pharmacokinetic, and ultradian endocrine models. These results suggest that BINODEs offer a useful compromise between mechanistic interpretability and data-driven flexibility for modeling partially known biochemical or biological dynamical systems.

Luis L. Fonseca, Reinhard C. Laubenbacher, Lucas Böttcher

Laboratory for Systems Medicine, Department of Medicine, University of Florida, Gainesville, FL, USA · Department of Computational Science and Philosophy, Frankfurt School of Finance and Management, Frankfurt am Main, 60322, Germany

Dynamical Systems Neural Processes Nonlinear System Identification Neural ODEs Ordinary Differential Equations

May 21, 2026

Holomorphic Neural ODEs with Kolmogorov-Arnold Networks for Interpretable Discovery of Complex Dynamics

Complex dynamical systems governed by holomorphic maps such as

z^2 + c

exhibit fractal boundaries with extreme sensitivity to initial conditions. Accurately modelling these structures from data requires methods that respect the underlying complex-analytic geometry, yet Multi-Layer Perceptrons (MLPs) within Neural Ordinary Differential Equations (Neural ODEs) lack complex-analytic priors, violate the Cauchy--Riemann conditions, and function as opaque approximators incapable of yielding governing equations. We introduce Holomorphic KAN-ODE, a framework that replaces the MLP with a Kolmogorov-Arnold Network (KAN) whose learnable B-spline activations reside on network edges, and incorporates Cauchy--Riemann equations as a differentiable regularization to preserve holomorphic structure. We evaluate on six families of complex dynamical systems spanning polynomial and transcendental classes. With only 280 parameters (

16\times

fewer than the MLP baseline), the network achieves velocity-field

R^2 > 0.95

on all six systems, correctly identifies all six governing symbolic families through automatic spline-to-formula fitting, and reconstructs Julia set fractal boundaries with up to 98.0% agreement. Crucially, the model exhibits only 4% MSE degradation under 10% observation noise versus

15.2\times

for MLPs, and achieves 90.4% improvement in transfer learning from quadratic to cubic dynamics. While the MLP attains lower pointwise reconstruction error due to its larger capacity, the KAN uniquely provides interpretable symbolic equations, enforced holomorphic structure, and superior noise resilience, capabilities that are entirely absent in black-box architectures. These results establish KANs as a parameter-efficient, interpretable alternative to MLPs for physics-informed discovery of holomorphic dynamics.

Bhaskar Ranjan Karn, Dinesh Kumar

ODE Discovery Complex-Valued Neural Networks Kolmogorov-Arnold Networks Nonlinear System Identification Chaotic Dynamical Systems Neural ODEs+1

May 19, 2026

Unlocking the Potential of Continual Model Merging: An ODE Perspective

Continual Model Merging (CMM) enables rapid customization of foundation models by sequentially incorporating task-adapted models without repeated retraining. However, existing merging rules usually update the deployed model through fixed algebraic or projection-based operations, providing limited control over how much previously accumulated knowledge should be retained relative to the incoming task model. This limitation leads to unstable retention and performance degradation in long task streams, and becomes more pronounced when tasks have heterogeneous utilities. We propose ODE-driven Merging (ODE-M), a controllable framework that formulates each continual merge as a trajectory in parameter space rather than a one-step endpoint update. Motivated by mode connectivity, ODE-M constructs a barrier-aware trajectory using a rectified time-dependent velocity field, where lightweight first-order feedback from a small calibration set suppresses loss-increasing motion while preserving progress toward the incoming model. The next merged model is then obtained by selecting an operating point along this trajectory through a utility-aware time schedule, providing an explicit mechanism for balancing retained historical knowledge and incoming task expertise. Extensive experiments on standard CMM benchmarks show that ODE-M consistently improves over strong continual merging baselines across CLIP ViT backbones, stream lengths, and heterogeneous task-utility settings.

Lihong Lin, Haidong Kang

Northeastern University, Shenyang, China

Mode Connectivity Ordinary Differential Equations Model Merging

May 13, 2026

MPINeuralODE: Multiple-Initial-Condition Physics-Informed Neural ODEs for Globally Consistent Dynamical System Learning

Neural ordinary differential equations (Neural ODEs) often fit training trajectories while generalizing poorly to unseen initial conditions and long horizons. We propose MPINeuralODE, which combines a soft physics-informed residual with a Multiple-Initial-Condition (MIC) multiple-shooting curriculum whose ingredients are structurally complementary: the physics term anchors the vector-field magnitude on the support that MIC enlarges. We evaluate along three axes: out-of-sample error, long-horizon stability, and Hamiltonian drift, which together expose whether the learned dynamics recover the underlying vector field. On Lotka-Volterra, MPINeuralODE achieves the lowest out-of-sample and long-horizon MSE among data-driven methods, with a 24% reduction over the baseline Neural ODE, while essentially matching the PINN ablation on Hamiltonian drift.

Lake Yang, Antonio Malpica-Morales, Frank Ioannis Papadakis Wood +1

Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom

Neural Network Generalization System Identification Physics-Informed ML Neural ODEs Ordinary Differential Equations

May 8, 2026

Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency

Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property, which we call Symmetry Rupture, serves two purposes. As a training regularizer, it confines the hypothesis space to flows that compose consistently across temporal scales. As an inference oracle, it lets the solver select the largest step size that preserves internal consistency, replacing the local truncation error that conventional adaptive solvers depend on. On the diffusion-reaction benchmark under time-informed inference, our method reduces rollout RMSE by 87% while using 5x fewer function evaluations than a Neural ODE baseline. In the more demanding direct auto-regressive setting, where the model must predict distant future frames without intermediate temporal cues, our adaptive solver allocates compute based on local geometric complexity -- maintaining the lowest rollout RMSE on two of three PDE benchmarks while baselines either diverge or require up to an order of magnitude more function evaluations to remain stable.

Yuxiang Luo, Andrew Perrault

Department of Computer Science and Engineering The Ohio State University

Neural PDE Solvers System Identification Consistency Regularization Ordinary Differential Equations

May 8, 2026

Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation

Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential equation modeling also requires incorporating domain knowledge to ensure physical plausibility. To address this gap, we propose DoLQ, a method for discovering ordinary differential equations with LLM-based qualitative and quantitative evaluation. DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations. Our code is available at https://github.com/Bon99yun/DoLQ.

Sum Kyun Song, Bong Gyun Shin, Jae Yong Lee

Department of Artificial Intelligence, Chung-Ang University, Seoul, Republic of Korea · Department of AI Big Data, Daejin University, Pocheon-si, Republic of Korea.

AI Agents for Scientific Discovery Autonomous Scientific Discovery Large Language Model-Guided Scientific Discovery ODE Discovery Symbolic Regression Ordinary Differential Equations

Apr 20, 2026

Physics-Informed Neural Networks: A Didactic Derivation of the Complete Training Cycle

This paper is a step-by-step, self-contained guide to the complete training cycle of a Physics-Informed Neural Network (PINN) -- a topic that existing tutorials and guides typically delegate to automatic differentiation libraries without exposing the underlying algebra. Using a first-order initial value problem with a known analytical solution as a running example, we walk through every stage of the process: forward propagation of both the network output and its temporal derivative, evaluation of a composite loss function built from the ODE residual and the initial condition, backpropagation of gradients -- with particular attention to the product rule that arises in hidden layers -- and a gradient descent parameter update. Every calculation is presented with explicit, verifiable numerical values using a 1-3-3-1 multilayer perceptron with two hidden layers and 22 trainable parameters. From these concrete examples, we derive general recursive formulas -- expressed as sensitivity propagation relations -- that extend the gradient computation to networks of arbitrary depth, and we connect these formulas to the automatic differentiation engines used in practice. The trained network is then validated against the exact solution, achieving a relative

L^2

error of

4.290 \times 10^{-4}

using only the physics-informed loss, without any data from the true solution. A companion Jupyter/PyTorch notebook reproduces every manual calculation and the full training pipeline, providing mutual validation between hand-derived and machine-computed gradients.

Abdeladhim Tahimi

Campus de Engenharias e Ciências Agrárias (CECA), Universidade Federal de Alagoas (UFAL), Rio Largo, Alagoas, 57100-000, Brazil

Backpropagation Ordinary Differential Equations Physics-Informed ML

Apr 17, 2026

Neuro-Symbolic ODE Discovery with Latent Grammar Flow

Understanding natural and engineered systems often relies on symbolic formulations, such as differential equations, which provide interpretability and transferability beyond black-box models. We introduce Latent Grammar Flow (LGF), a neuro-symbolic generative framework for discovering ordinary differential equations from data. LGF embeds equations as grammar-based representations into a discrete latent space and forces semantically similar equations to be positioned closer together with a behavioural loss. Then, a discrete flow model guides the sampling process to recursively generate candidate equations that best fit the observed data. Domain knowledge and constraints, such as stability, can be either embedded into the rules or used as conditional predictors.

Karin Yu, Eleni Chatzi, Georgios Kissas

Institute of Structural Engineering, ETH Zurich, Zurich, Switzerland · ETH AI Center, ETH Zurich, Zurich, Switzerland · Swiss Data Science Center, Zurich, Switzerland

Autonomous Scientific Discovery Neurosymbolic Learning ODE Discovery Symbolic Regression Ordinary Differential Equations

Mar 12, 2026

Inverse Neural Operator for ODE Parameter Optimization

We propose the Inverse Neural Operator (INO), a two-stage framework for recovering hidden ODE parameters from sparse, partial observations. In Stage 1, a Conditional Fourier Neural Operator (C-FNO) with cross-attention learns a differentiable surrogate that reconstructs full ODE trajectories from arbitrary sparse inputs, suppressing high-frequency artifacts via spectral regularization. In Stage 2, an Amortized Drifting Model (ADM) learns a kernel-weighted velocity field in parameter space, transporting random parameter initializations toward the ground truth without backpropagating through the surrogate, avoiding the Jacobian instabilities that afflict gradient-based inversion in stiff regimes. Experiments on a real-world stiff atmospheric chemistry benchmark (POLLU, 25 parameters) and a synthetic Gene Regulatory Network (GRN, 40 parameters) show that INO outperforms gradient-based and amortized baselines in parameter recovery accuracy while requiring only 0.23s inference time, a 487x speedup over iterative gradient descent.

Zhi-Song Liu, Wenqing Peng, Helmi Toropainen +5

Department of Computational Engineering, Lappeenranta-Lahti University of Technology, Lahti, Finland · Atmospheric Modelling Center Lahti (AMC-Lahti), Finland · University of Helsinki, Helsinki, Finland +3

Parameter Estimation Ordinary Differential Equations Neural Operators Learning to Optimize Neural Surrogate Modeling

Mar 11, 2026

MCMC Informed Neural Emulators for Uncertainty Quantification in Dynamical Systems

Neural networks are a commonly used approach to replace physical models with computationally cheap surrogates. Parametric uncertainty quantification can be included in training, assuming that an accurate prior distribution of the model parameters is available. Here we study the common opposite situation, where direct screening or random sampling of model parameters leads to exhaustive training times and evaluations at unphysical parameter values. Our solution is to decouple uncertainty quantification from network architecture. Instead of sampling network weights, we introduce the model-parameter distribution as an input to network training via Markov chain Monte Carlo (MCMC). In this way, the surrogate achieves the same uncertainty quantification as the underlying physical model, but with substantially reduced computation time. The approach is fully agnostic with respect to the neural network choice. In our examples, we present a quantile emulator for prediction and a novel autoencoder-based ODE network emulator that can flexibly estimate different trajectory paths corresponding to different ODE model parameters. Moreover, we present a mathematical analysis that provides a transparent way to relate potential performance loss to measurable distribution mismatch.

Heikki Haario, Zhi-Song Liu, Martin Simon +1

Lappeenranta-Lahti University of Technology LUT, Lahti, Finland · Frankfurt University of Applied Sciences, Frankfurt am Main, Germany · University of Huddersfield, Huddersfield, United Kingdom

Ordinary Differential Equations Uncertainty Quantification Neural Surrogate Modeling Dynamical Systems

Mar 2, 2026

Markovian ODE-guided scoring can assess the quality of offline reasoning traces in language models

Reasoning traces produced by generative language models are increasingly used for tasks ranging from mathematical problem solving to automated fact checking. However, existing evaluation methods remain largely mechanical and fail to capture human-centric notions of reasoning quality in a way that generalizes across varied and progressively degraded reasoning. We introduce MarODE, an offline evaluation framework that assigns quality scores to reasoning traces. Its effectiveness is assessed using human-centric perturbations and human judgments, which jointly evaluate the fundamental dimensions of an evaluation metric - goodness and soundness. The approach is grounded in a Markovian formulation of reasoning progression and an ordinary differential equation based characterization of trace dynamics, enabling efficient evaluation of reasoning quality. In a large-scale evaluation, MarODE outperforms existing baselines by over 250% under Somers' D correlation. Our results emphasize the value of theory-driven evaluation frameworks as reasoning traces become central to language model-based systems.

Arghodeep Nandi, Ojasva Saxena, Tanmoy Chakraborty

Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi, New Delhi, India · Indian Institute of Technology Delhi, Abu Dhabi, UAE

Ordinary Differential Equations Language Model Reasoning Evaluation Language Model Generation Evaluation Reasoning Trace Analysis