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.

Introducing SINFONIA: Symplectic, slimplectic and Magnusian (Neural) Flows for Orbital Numerical Integration and Acceleration

Long-duration gravitational-wave modelling must resolve fast orbital motion together with slow dissipative evolution while preventing small numerical errors from accumulating into secular phase drift. Here we ask whether the finite-time evolution map itself can be learned as an explicit, differentiable, structure-preserving object and then repeatedly composed through a complete inspiral. We construct three neural-flow architectures: a symplectic and slimplectic flow on Galley's doubled phase space, [SINFONIA-J0]; a Taylor-anchored flow, [SINFONIA-J1]; and a Magnusian flow that learns the finite-time dissipative correction in the interaction picture, [SINFONIA-J2]. Applied to a 2.5PN neutron-star inspiral, all three expose the same controlling mechanism: long-time accuracy is governed not by pointwise map error alone, but by its signed projection onto a single secular channel fixed by energy--angular-momentum balance. Encoding this structure allows the learned maps to remain accurate through $10^{2}$--$10^{5}$ window compositions to coalescence at timesteps of a full orbital period and beyond, reaching chained phase errors orders of magnitude below a benchmark slimplectic integrator at lower cost. The same secular structure can also be exploited for physics inference: when the channel is left unconstrained, the accumulated phase retains enough information to recover an un-modelled dynamical-friction-like force, both parametrically and as a learned function of separation. Network-off controls isolate the contribution of learning from the analytic structure already built into each map. These results establish a proof of concept for structure-preserving learned evolution maps as tools for fast long-duration integration and physics inference in gravitational-wave source modelling.

Learning Reduced-Order Dynamics with Singularity via Latent-Augmented Neural Ordinary Differential Equations

This paper addresses the issue of self-intersecting trajectories (in phase space) in industrial reduced-order modeling and proposes the Latent-Augmented Neural Ordinary Differential Equations (LA-NODEs) framework. From the perspective of artificial intelligence, the proposed method augments conventional neural ordinary differential equations to enhance model expressiveness, enabling the representation of conflicting vector fields that may arise in reduced-order systems, thereby improving learning accuracy. Through theoretical analysis, the underlying mechanism of the framework is established, and a condition for determining the minimum required augmentation dimension is derived. From the perspective of engineering applications, the effectiveness of the proposed method is validated on the reduced-order system of two representative industrial models, namely an interior permanent magnet synchronous motor (IPMSM) drive and a distributed energy system (DES). Experimental results demonstrate that the proposed method can recover system features that are difficult to capture using conventional approaches and achieve superior performance in terms of prediction accuracy and modeling fidelity, thereby providing an effective approach for high-precision data-driven modeling of complex industrial systems.

Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies

We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.

A matched-integrator evaluation of Hamiltonian neural networks on pendulum and Kepler dynamics

Hamiltonian Neural Networks (HNNs) parameterize conservative dynamics through a learned scalar Hamiltonian, providing an architectural prior that is absent from generic vector-field neural networks. We evaluate this prior under a controlled protocol in which an HNN and a parameter-matched feedforward baseline are trained on the same RK4-generated trajectories, use the same central-difference derivative targets and optimization settings, and are integrated at inference with the same RK4 scheme. Results are reported over five independent training seeds. On the nonlinear pendulum, the HNN reduces mean energy drift by 42-fold and mean trajectory MSE by 15.8-fold at T = 100, approximately 16 pendulum periods. Its energy drift also remains bounded and exhibits substantially lower seed-to-seed variability than the standard-network baseline. An energy-stratified analysis shows that the difference becomes more pronounced as trajectories explore more nonlinear regions of phase space. As an additional diagnostic, we examine an explicit Störmer--Verlet-style rollout of the learned HNN. Because the learned Hamiltonian is not constrained to the separable form H(q,p) = T(p) + V(q), the standard symplecticity guarantee of velocity Verlet does not directly apply. We further apply the same matched-integrator protocol to the three-dimensional Kepler two-body problem. The HNN again exhibits lower trajectory, energy, and angular-momentum drift than the parameter-matched baseline. These experiments provide a controlled study of how Hamiltonian parameterization affects long-horizon prediction and physical consistency across two conservative dynamical systems.

Training Crossroads for Recurrent Vision Transformers: Recurrence, Neural ODEs, and Deep Supervision

Vision Transformers (ViTs) achieve strong image-recognition performance, but their parameter count grows linearly with depth when each block is independently parameterized. Single-block recurrent ViTs (bViT) remove this growth by repeatedly applying one shared block. Rather than proposing a new architecture, we fix a bViT and provide a controlled empirical characterization of three training and inference regimes under a common CIFAR-100 protocol, asking: (i)~when does recurrence beat independently parameterized depth---at matched FLOPs or at matched parameter memory? (ii)~when a residual recurrent block is trained through an ODE solver, does solver order act as numerical refinement or as an architectural bias? and (iii)~what does robustness beyond the training horizon cost in nominal accuracy? We find that standard ViTs remain preferable when FLOPs are the primary constraint, whereas recurrent ViTs offer a better accuracy--parameter trade-off under memory constraints. Consistent with the standard view of residual networks as Euler discretizations of ODEs, the continuous-time analogue of a residual recurrent block is the state-subtracted vector field $\dot{z}=F_θ(z)-z$; although known in principle, this distinction is easy to violate when the block is wrapped as a black-box vector field, and we qualify the cost at few accuracy points. Because the vector field is learned jointly with the solver, higher-order solvers act as a solver-induced architectural bias rather than a numerical-accuracy improvement, and their gains are not uniform. Finally, stage-wise deep supervision traces an accuracy--robustness frontier: it does not improve nominal accuracy, but degrades gracefully far beyond the training horizon, where naive recurrence collapses to near-random performance.

Spatio-Temporal Prediction of Unsteady Airfoil Aerodynamics Using Augmented Graph Neural Ordinary Differential Equations with Exogenous Controls

Unsteady aerodynamic phenomena, such as gusts, turbulence, and fluid-structure interactions affect an aircraft during flight. For design, optimisation and certification, it is indispensable to quantify such unsteady aerodynamic effects. Industry-standard computational fluid dynamics methods, such as solving the unsteady Reynolds-averaged Navier-Stokes equations or the linearized frequency domain method, are either computationally expensive or restricted by assumptions like linearity. Once trained, machine learning methods are capable of computing non-linear relationships very fast, making them suitable as surrogate models. By autoregressively applying graph neural networks (GNNs), operating on a discretised spatial domain, spatio-temporal predictions can be made. However, autoregressive GNNs suffer from error accumulation leading to unstable rollouts over time. Here we show that combining GNNs with augmented Neural Ordinary Differential Equations yields temporally stable predictions of the surface forces on a pitching airfoil. We found that our approach, called GNODE, based on Graph Neural Ordinary Differential Equations, provides temporally more stable, spatially smoother, and overall more accurate results than an autoregressive GNN baseline. Tests are conducted on a dataset consisting of a simulations of a pitching airfoil, including transonic shocks, transient behaviour and dynamic non-linearities. Augmenting GNODEs with additional latent dimensions improves the expressivity and accuracy by capturing underlying history effects. The developed method demonstrates an approach that is suitable to model non-linear spatio-temporal systems with exogenous inputs.

Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$$ The threshold $q = 1$ is established via two independent arguments. Classical ODE theory gives a global forward flow on $[0, T]$ at $q \le 1$ and exhibits divergent velocity fields at any $q > 1$. The optimal-control analysis, via the Hamilton-Jacobi-Bellman equation, sharpens this to a selection statement: the training optimum is bang-bang on the boundary of the admissible class, so the optimum at $q > 1$ blows up while the optimum at $q \le 1$ is safe by construction. The exponent criterion $q \le 1$ is thereby a necessary and sufficient condition for stable training. It clarifies architectural placements that ensure the stability of training and inference, explaining, for instance, the stabilizing role of layer normalization. The sublinear-growth velocity fields form \emph{the right function space} on which forward dynamics, adjoint sensitivity, and architectural composition are all well-controlled. An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k \le 1$ at the level of architectural primitives, in place of ad hoc trial and error in the search for stable neural architectural designs. A parameter-free modification reduces the supercritical Mamba block from $q = 5$ to $q = 1$ without layer normalization, demonstrating this point. Experiments on Mamba and PatchTST confirm that the $q \le 1$ variants train stably: the criterion is the input-magnitude exponent, not the presence of a normalization layer.

Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence

Graph Neural Differential Equations (GNDEs) model continuous-time graph dynamics by parameterizing Neural ODE velocity fields with Graph Neural Networks. Their local, size-independent filters suggest a zero-shot size-transfer principle: train on a small graph and deploy on larger, similar graphs without retraining. We develop a quantitative theory for this principle on sparse random graphs sampled from graphons. We consider Graphon Neural Differential Equations (Graphon-NDEs) and adjoint Graphon-NDEs as the infinite-node limits of the forward and adjoint GNDE systems, and establish well-posedness. For an $n$-node random graph with sparsity parameter $α_n$, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate $O((α_n n)^{-1/2})$, up to logarithmic factors, with high probability. We also establish uniform-in-time convergence bounds for adjoint systems governing hidden-state and parameter gradients. We further study discretize-then-optimize (DTO) and optimize-then-discretize (OTD) training. Under explicit Euler discretization with $M$ steps, we show that DTO and OTD are asymptotically consistent, with hidden-state and local parameter-gradient discrepancies of orders $O(1/M)$ and $O(1/M^2)$, respectively, up to sparsity and logarithmic factors. Experiments on HSBM and tent graphons support the theoretical rates, while zero-shot transfer experiments across four graphon classes demonstrate accurate deployment of learned GNDEs on larger independently sampled graphs.

Frequency-Domain Neural ODEs for Modeling Non-Linear Dynamical Systems

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

Learning Hybrid Biophysical Neuron Models with Neural ODEs

Biophysical neuron models link measurements of neural activity to underlying cellular mechanisms. Yet, a central challenge is that the kinetics of many ion channels are poorly characterized, and practical simplifications -- omitting channels or reducing morphological detail -- introduce systematic gaps between model and biology. Bridging these gaps requires approaches that can flexibly discover unmodeled dynamics while preserving mechanistic interpretability. Here, we introduce a hybrid modeling framework that embeds neural ordinary differential equations into conductance-based biophysical models to capture unknown currents or mis-specified channel kinetics. By parameterizing the neural ODE in terms of voltage-dependent steady-state and time-constant functions, we recover interpretable gating dynamics directly from voltage recordings without assuming a functional form. We show that the hybrid model fits the gating kinetics of 2400 ion channel models and recovers unknown gating dynamics from single current-clamp recordings, generalizing to out-of-distribution stimulus regimes under realistic inputs and parameter misspecification. We also use our method to reduce a multicompartment model of a cortical neuron into a single-compartment hybrid model with a learned axial current, yielding up to an order of magnitude lower computational cost. Together, our results establish a plug-and-play framework for selectively replacing unknown components of conductance-based models with neural ODEs while preserving their mechanistic structure.

Theory of learning of high-dimensional controlled non-linear dynamical systems (I): models and methods

Neural ordinary differential equations (neural ODEs) have rapidly gained prominence as a powerful and unifying framework for conceptualizing artificial neural networks, elegantly connecting the continuous-time modeling of dynamical systems with the discrete, data-driven paradigm of modern deep learning. Beyond their practical advantages they offer fresh theoretical insights into the training and generalization properties of neural networks. The distinctive feature of this framework is its dual dynamical nature: inference dynamics, which govern the ODE evolution during forward computation, and training dynamics, which control the optimization of model parameters. This makes neural ODEs a particularly well-suited theoretical framework for studying a large variety of settings such as multi-layer neural networks (ResNets for example), autoregressive models (with next-token generation dynamics), generative models, and recurrent neural networks in theoretical neuroscience. In this work, we introduce a theoretically grounded class of models for studying neural ODEs trained via online stochastic gradient descent. We solve the training dynamics of these models via dynamical mean field theory and derive learning curves in the high-dimensional limit.

Learning Chaotic Dynamics through Second-Order Geometric Supervision

Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus match values and tangents at the supervised states yet curve differently from the truth, remaining locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that enforcing second-order consistency mitigates these failures, but forming the full Hessian is prohibitive in high dimensions. We propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the nominal Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at $\mathcal{O}(d^2)$ cost without forming the $\mathcal{O}(d^3)$ Hessian tensor. Using only Jacobian evaluations, the method scales to high dimensions where explicit Hessian matching does not. Numerical experiments confirm that second-order methods are robust. For Lorenz~63, first-order methods produce catastrophic Lyapunov-exponent outliers under minimal temporal supervision, which second-order methods eliminate while recovering the correct attractor. For coupled Lorenz~96, an out-of-distribution forcing sweep separates the methods: all agree up to $F=16$, but beyond $F=18$ only second-order methods preserve the invariant measure and Lyapunov spectrum. On both systems, randomized Jacobian matching performs comparably to explicit Hessian matching at much lower cost.

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.