Increased computing power has transformed computational science and improved predictive capabilities for complex systems, but many phenomena still exceed current simulation capabilities or require rapid evaluations for fast design iterations. Closure models address this challenge by approximating unresolved effects in reduced or incomplete dynamical systems. Data-driven closures offer a flexible mechanism for improving model accuracy and applicability, but approaches that embed the closure within a computational model for construction and training phases can be computationally expensive and cause numerical instabilities in the computational model during training. To avoid the challenges posed by embedded approaches, staged approaches to closure-model learning have been developed. The first stage uses data assimilation to infer state and closure trajectories from observations. The second stage trains a machine learning closure model offline using the assimilated data, thereby avoiding repeated simulator calls during training. In this work, we investigate the cost and accuracy tradeoffs between embedded and staged data-driven closure modeling approaches for a synthetic epidemiological SIQR model with a nonlinear state-dependent quarantine closure. The results illustrate a practical accuracy-efficiency trade-off: for this exemplar, embedded training provides the strongest average accuracy, while the staged approach yields comparable qualitative behavior with a moderate increase in error, substantially fewer model evaluations, and improved optimization robustness.
Many control tasks require driving the system into a prescribed goal set and subsequently keeping it there. The considered setting assumes that a working controller is available that provably reaches and maintains the goal, albeit with suboptimal performance. This controller is referred to as the fallback policy. Conversely, although state-of-the-art reinforcement-learning (RL) algorithms can discover policies with impressive empirical performance, the neural networks at their core act as opaque black boxes learned through data-driven exploration and, as such, provide no theoretical mechanism to guarantee that the system will ever reach---and stay within---the prescribed goal set. This article presents CALF-Wrapper, a method that combines the fallback policy with a pretrained RL base policy and uses the critic estimate associated with the base policy to select between them during deployment. The theoretical analysis establishes that CALF-Wrapper inherits the goal-reaching property of the fallback policy under the stated assumptions and provides probability bounds for the settling time and the maximum distance from the goal set over a prescribed time interval. The experimental study evaluates CALF-Wrapper in four simulated control environments of different complexity. Comparisons include the base policy, fallback policy, and constrained reinforcement-learning methods formulated for the same objective. Sensitivity and ablation studies provide direct hyperparameter recommendations. The experiments show how CALF-Wrapper can provide effective task performance while attaining the maximum possible observed goal-reaching rate. CALF-Wrapper requires no additional dynamics model, online learning, or constrained optimization during execution.
We develop nonreciprocal spherical attention as a mathematical framework for coarsening and compressed nearest-neighbor retrieval. For two shrinking acute blocks whose centers have zero skew pairing and whose total mass ratio is subexponential in inverse temperature, we prove $\lim_{\beta\to\infty}\beta^{-1}\log T_\beta=1-\cos\phi$, without restricting particle-count growth. An explicit three-body flow has two rigorously separated metastable exponents. For finite compression, mass centering yields a sharp $O(r^2)$ attention-output error bound at fixed temperature, with an explicit spherical-normalization correction. We formulate skew query-side score calibration as a strictly convex moment problem and derive a unique normal equation, residual-based optimization certificates, and bounded low-rank projections. A norm-aware margin objective calibrates native squared-distance rankings. Experiments use semantic retrieval data, one million visual descriptors with three codebook seeds, and a ten-million-vector FAISS IVF-PQ index with recomputed exact cosine ground truth. On SIFT1M, skew margin calibration raises native-L2 hit@10 from 0.8926 to 0.8948; symmetric calibration reaches 0.8996. The ten-million-vector index reaches 0.8827 hit@10 at 0.1794 ms per query in a four-thread batch benchmark, while validation selects no skew update. Exact streamed attention over ten million keys evaluates compressed memory on 1,024 held-out queries. Independent matrix solves, automatic differentiation, finite differences, interval arithmetic, full-flow tests, and nine Rocq-checked algebra lemmas support reproducibility. Directionality alone does not establish retrieval superiority.
Many systems are observed only through unpaired snapshots, with population samples available at each time point but no correspondence between samples at different times. These snapshots determine the one-time distributions but leave the temporal coupling unspecified, requiring a modeling prior to connect successive times. We propose the lift-project stochastic differential equation (SDE), an autonomous SDE on the observed coordinate and a few hidden coordinates, trained only to match the snapshots. Most existing methods, including time-dependent SDEs, Schr\"odinger bridges and optimal transport, evolve the observed coordinate alone. For a scalar observation, the classical comparison theorem implies that all such models have non-negative lagged covariance, so on data whose lagged covariance turns negative, their error has a computable floor. The lift-project SDE is not bound by this floor. On two representative partially observed systems, the Duffing and pendulum oscillators, at comparable fit to the snapshots, the proposed method reduces the lagged-covariance error by factors of 2.4 and 6.6 relative to the strongest baseline and is the only model below the error floor; one hidden coordinate is already enough for such improvement. Furthermore, if a second experiment changes only the initial state of the hidden coordinate, no single model that evolves the observed coordinate alone can fit both experiments, whereas a single lift-project SDE can.
Analog computers solve problems through physical representations of mathematical variables and relations, often using the dynamics of electronic circuits and their associated signals. Many analog computers directly represent nonlinear dynamical systems; they encode their solutions in the signals that they generate. Although most modern scientific computing relies on digital architectures with inherently discrete-valued representations and algorithmic approaches, the design constraints of such architectures increasingly limit both performance and energy efficiency. Analog systems can circumvent some of these constraints; they execute specific computations very efficiently by mapping them directly onto physical processes using smaller and simpler computing elements than the ones in digital architectures. In this article, we provide an interdisciplinary perspective on modern analog and hybrid (i.e., analog-digital) computing. We discuss how modern scientific computing workflows can leverage analog approaches in various applications, such as in simulating dynamical systems, performing optimization, and enabling more energy-efficient implementations of artificial intelligence.
We develop a divergence-kernel method for jointly optimizing neural SDE drift and diffusion fields and the initial Gaussian distribution, using losses that depend directly on the terminal density. In two- and eight-dimensional Gaussian-mixture experiments, we learn time-independent fields and obtain far fewer off-mode samples than score-based generators with similar parameter counts. The drift attracts samples toward the modes, while the diffusion helps disperse excess probability between them.
A nonlinear dynamical system can be characterized by the autonomous evolution of its state in time and its dependence on system parameters. As a parameter varies, the dynamics can change quantitatively and, at bifurcation points, qualitatively. When the governing equations are unknown, a data-driven model should therefore capture both elements from observational data, rather than reproduce the dynamics at a single operating condition only. Parameter-aware reservoir computing (PARC) is a model-free framework for this purpose, where the parameter is supplied to a reservoir computer through an additional input channel. PARC is trained on time series recorded at several parameter values, and the trained reservoir, run in closed loop as an autonomous dynamical system, generates trajectories at parameter values that were never observed. rc_prediction is an open-source Python package that implements this workflow. It provides multi-parameter training of a reservoir, autonomous closed-loop prediction at user-specified parameter values, parameter scans, and a standard echo state network for forecasting at a fixed parameter. Hyperparameter optimization and optional GPU acceleration through PyTorch are also provided. We use three benchmark dynamical systems as examples to demonstrate its usage. As one of the featured applications, here we illustrate the prediction of critical transitions and system collapse: trained only on pre-critical data, the model is queried beyond the training range to predict whether the system collapses, to estimate the critical parameter $p_c^{*}$, and to characterize transient lifetimes.
Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like $\sqrt{{\log(\beta+1)}/{\beta}}\exp(Ct)+\exp(-ct)$ in terms of the temperature parameter $\beta^{-1}\to 0$ and inference time $t\geq 0$. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as $t\to\infty$, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order $\log\beta$, which correspond to the early-phase behavior most relevant for practical finite-depth models, the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite $\beta$ and large $t$ the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.
We develop a unified matrix-spectral framework for analyzing stability and interpretability in deep neural networks. Representing networks as data-dependent products of linear operators reveals spectral quantities governing sensitivity to input perturbations, label noise, and training dynamics. We introduce a Global Matrix Stability Index that aggregates spectral information from Jacobians, parameter gradients, Neural Tangent Kernel operators, and loss Hessians into a single stability scale controlling forward sensitivity, attribution robustness, and optimization conditioning. We further show that spectral entropy refines classical operator-norm bounds by capturing typical, rather than purely worst-case, sensitivity. These quantities yield computable diagnostics and stability-oriented regularization principles. Synthetic experiments and controlled studies on MNIST, CIFAR-10, and CIFAR-100 confirm that modest spectral regularization substantially improves attribution stability even when global spectral summaries change little. The results establish a precise connection between spectral concentration and analytic stability, providing practical guidance for robustness-aware model design and training.
We develop a finite-dimensional sensitivity framework for studying stability in learning systems whose states include representations, parameters, and update variables. The central object is the \emph{Learning Stability Profile}, a collection of directional sensitivity operators that records how perturbations in inputs, parameter initialization, and update mechanisms propagate along a specified learning trajectory. The main result is a Lyapunov criterion for controlling this profile. Under explicit regularity, coercivity, and dissipation assumptions, an incremental Lyapunov energy yields uniform or exponentially decaying bounds on the associated linearized transition operators. The result is stated as a sufficient stability criterion, not as an unconditional converse theorem. The framework also distinguishes terminal decay, profile-wise boundedness, and subexponential growth, avoiding the identification of nonpositive growth exponents with uniform boundedness. The profile is then specialized to several standard learning mechanisms. Spectral bounds give forward sensitivity estimates for feedforward networks. Dissipativity and step-size restrictions give stability bounds for residual architectures. Mean-square contraction assumptions yield parameter and update sensitivity bounds for stochastic gradient methods. Locally Lipschitz systems, including piecewise-linear networks, proximal maps, projected updates, and recurrent or state-space recursions, are handled through Clarke generalized Jacobians and variational Lyapunov inequalities. The resulting framework provides a common stability language for architecture, optimization, stochasticity, and nonsmoothness. Its role is structural: it organizes known stability mechanisms within one perturbation calculus while keeping the hypotheses needed for each guarantee explicit.
For a time-homogeneous Markov process, the fixed-lag solution operator \( (T_\tau f)(x)=\mathbb E[f(X_{t+\tau})\mid X_t=x] \) 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.