We construct, for each deterministic Turing machine that halts on all inputs, one polynomial vector field with rational coefficients that performs the same logical computation at unit Euler step, at every smaller or variable step in $(0,1]$ with divergent sum, and along the flow. This unifies discrete-time and continuous-time computation by polynomial dynamics, which earlier characterizations treated separately. For a machine making $T$ transitions in space $H$, a logical scale $h=2^{-s}$ with $s=O(H)$ suffices, with physical time $O(s(T+s))$, drift disturbances up to $h^{2K}$ for a fixed exponent $K$, and $2Ks+3$ fractional bits at unit step. On a fixed compact domain, polynomial time and robust eventual decision characterize P and PSPACE, respectively, with polynomial scale thresholds and no supplied stabilization horizon in the space case. Without sign, floor or conditional primitives, margin amplification at bounded speed, overlapping polynomial gates and interval-product estimates give the common realization, from which we derive shared logical traces, fixed-field PSPACE-complete prediction and a robust safety family with polynomial certificates common to all Euler steps and the flow.
For a time-homogeneous Markov process, the fixed-lag solution operator \( (T_τf)(x)=\mathbb E[f(X_{t+τ})\mid X_t=x] \) of the Kolmogorov backward equation evolves observables linearly and deterministically, even when the dynamics are stochastic. We approximate this operator using a two-stage method where we first approximate this operator directly from data, without knowledge of the governing equation (e.g., Fourier Neural Operator), and then compress the Galerkin projection onto a finite-dimensional observable space that is fixed in advance or selected variationally. The error analysis traces the learning error of the estimator, the sampling error of the Galerkin inner products, and the invariance and reconstruction properties of the observable space to the computed spectra and forecasts. The sampling error enters only through a multiplicative factor. Near an isolated spectral cluster, the invariance defect of a subspace controls its distance from the target invariant subspace. The forecast error is controlled by the one-step error of the reduced model and the projection error of the observable; when the measure is invariant and the reduced model preserves constants and is stable on mean-zero observables, the bound is uniform in the horizon. For deterministic dynamics, a learning rate for Fourier neural operators (FNOs) gives one explicit bound for the learning error. Numerical experiments illustrate spectral approximation and forecasting, including long-rollout stability and a comparison with extended dynamic mode decomposition (EDMD) when fewer transition pairs than basis functions are observed.
John S. McAlister, Nina H. Fefferman, Tadele A. Mengesha
Coordination games with explicit spatial or relational structure are of interest to economists, ecologists, sociologists, and others studying emergent global properties in collective behavior. When assemblies of individuals seek to coordinate action with one another through myopic best response or other replicator dynamics, the resulting dynamical system can exhibit many rich behaviors. Here we seek to understand these coordination dynamics in the non-atomic setting by use of a nonlinear, nonlocal diffusion equation framework. For this class of nonlocal equations, we prove existence and uniqueness for the initial value problem in the free boundary case and, with certain conditions on the underlying game theory, we go further and prove a maximum principle, weak regularity results, and a characterization of asymptotic behavior. Additionally, with the help of several numerical results, we present several modeling results, characterizing stationary solutions both rigorously and through numerical experiments.
Yubin Lu, Xiaofan Li, Chun Liu, Qi Tang, Yiwei Wang
Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.
James Hazelden, Laura Driscoll, Eli Shlizerman, Eric Shea-Brown
In training a neural network with gradient descent (GD), each iteration induces a linear operator that governs the first-order updates to a model's internal state variables. We call this operator the Global Empirical Neural Tangent Kernel (GeNTK). In finite-width networks, GeNTK is typically intractable to form, leading prior work to focus on restrictive settings, such as tracking only outputs or infinite-width limits. Here, we bridge this gap by studying the structure of GeNTK across many models. Formulating the model state as the solution to a global implicit constraint, we derive GeNTK as a product of two operators: K, accounting for immediate parameter-to-state interactions, and P, describing internal state-to-state dependencies. For a broad class of "weight-based models", such as RNNs and transformers, we prove a universal Kronecker-core theorem showing that K admits an exact, forward-pass-computable form given by the Gram matrix of weight-site variables. This core gives a measurable state-dependent restriction on Euclidean GD, termed "self-referential bias": current activity filters which adjoint directions can drive a first-order update. For recurrent models (GRUs and RNNs), we examine the spectrum of GeNTK and formalize notions of low-rank, biased learning in terms of reduced spatial or temporal ranks of the full operator. Practically, this explains why some RNN initializations yield better learning, while some plateau, by biasing the GeNTK and thus restricting learning to a small domain of learnable tasks. To demonstrate computational viability, we validate the main theorems in a self-attention transformer and a pretrained Pythia transformer language model with 410 million parameters. Overall, our results expose tractable local rank and mode-selective biases in the finite-width GD geometry and open the door to principled study of the GeNTK, a bridge between theory and practice.
The curse of dimensionality associated with solving the Hamilton-Jacobi-Bellman equation for the value function remains a key obstacle for real-time optimal feedback control of high-dimensional nonlinear dynamical systems. This is especially severe in aerospace applications such as relaminarizing unsteady turbulent flows and recovering from aerodynamic stall where nonlinear dynamics play a significant role. We introduce Balanced Active Subspaces for Optimal Control (BASOC), a model reduction framework which seeks to find directions in state space along which the value function is most sensitive, and the distribution of states exhibits maximum variance along optimal trajectories. BASOC leverages trajectory optimization and the Pontryagin minimum principle to obtain state and co-state (value function gradient) data along locally optimal trajectories for constructing state and gradient covariance matrices. Balancing these matrices provides an oblique projection for approximating the value function, which can also be used to construct reduced-order models and surrogate feedback control laws. We show that for linear-quadratic optimal control problems, BASOC becomes equivalent to a snapshot method for initial condition balancing of the optimally controlled closed-loop realization. The framework is demonstrated on a nonlinear three-dimensional system where POD and balanced truncation are known to perform poorly, and on the nonlinear complex Ginzburg-Landau equation to demonstrate feasibility for larger-scale systems.
Eric Parish, Anthony Gruber, Patrick Blonigan, Irina Tezaur
We propose neural network operator inference (NN-OpInf): a structure-preserving, composable, and minimally restrictive operator inference framework for the non-intrusive reduced-order modeling of dynamical systems. The approach learns latent dynamics from snapshot data, enforcing local operator structure such as skew-symmetry, (semi-)positive definiteness, and gradient preservation, while also reflecting complex dynamics by supporting additive compositions of heterogeneous operators. We present practical training strategies and analyze computational costs relative to linear and quadratic polynomial OpInf (P-OpInf). Numerical experiments across several nonlinear and parametric problems demonstrate improved accuracy, stability, and robustness over P-OpInf and prior NN-ROM formulations, particularly when the dynamics are not well represented by polynomial models. These results suggest that NN-OpInf can serve as an effective drop-in replacement for P-OpInf when the dynamics to be modeled contain non-polynomial nonlinearities, offering potential gains in accuracy and out-of-distribution performance at the expense of higher training computational costs and a more difficult, non-convex learning problem.
Invariant manifolds and foliations associated to invariant pseudo-hyperbolic reference geometries have been well-studied in the context of dynamics implemented by a diffeomorphism. In this paper, we provide sufficient conditions for the existence of maximally smooth, \emph{locally} invariant manifolds and foliations containing a given \emph{locally} invariant, pseudo-hyperbolic reference geometry, in the context of dynamics implemented by a smooth map which need not be invertible. The theory is motivated by applications to the developing theory of large-step optimisation.
Let $G$ be a group, let $\mathbb{K}$ be a field, and let $V$ be a $\mathbb{K}$-vector space. We prove that there exists a bijective linear cellular automaton $V^G\to V^G$ whose inverse is not a cellular automaton if and only if $G$ is not locally finite and $\dim_{\mathbb{K}}V \geq |\mathbb{K}|^{\aleph_0}$. In particular, $|\mathbb{K}|^{\aleph_0}$ is the exact dimension threshold for non-reversibility. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.
Shu Zhou, K. Y. Michael Wong, Juntao Wang, David Shui Wing Hui, Daniel Ebler, Jie Sun
Recently proposed analog solvers based on dynamical systems, such as Ising machines, are promising platforms for large-scale combinatorial optimization. Yet, given the heuristic nature of the field, there is very limited insight on optimality guarantees of the solvers, as well as how parameter schedules shape dynamics and outcomes. Here, we develop a dynamical mean-field framework to analyze Ising-machine dynamics for finding the ground state energy of the Sherrington-Kirkpatrick(SK) model of spin glasses and identify mechanisms that enable rapid convergence to provenly near-optimal energies. For a fixed target energy density Ec, we show that solutions are typically reached within O(1) matrix vector multiplications, indicating constant time complexity. We further delineate theoretical limitations arising from different parameter-scheduling trajectories and demonstrate a pronounced benefit of temperature-only annealing for the Coherent Ising Machine. Building on these insights, we propose a general framework for designing optimized parameter schedules, thereby improving the practical effectiveness of Ising machines for complex optimization tasks. The superior performance of the dynamical solvers is illustrated by the attainment of the ground state energy of the SK model.
Predicting dynamics across bifurcations by extrapolating trained models is crucial in diverse fields of physics. Generalized synchronization provides a framework for understanding dynamical reconstruction by neural networks, especially in the context of reservoir computing. Building on this framework, we derive a formula combining pretrained readouts to reconstruct unseen chaotic dynamics. Using classical Lagrange interpolation theory, we derive rigorous bounds on readout errors and conjugacy residuals. Numerical examples demonstrate accurate prediction of chaotic dynamics across bifurcations.