Over the past decade, a growing body of work has established that implicit biases in learning dynamics fundamentally shape the solutions found by neural networks, governing their learned features and generalisation performance. However, despite these strides in deep learning theory, far less is known about representation learning in generative models based on dynamical systems, from normalising flows to diffusion. Importantly, empirical evidence suggests that continuous-time generative models can be prone to mode collapse, yet a learning theory explaining such biases remains elusive. To address this, we use tools from dynamical systems and operator theory to express the gradient flow of a general class of stochastic differential equations as a composition of linear operators, allowing us to relate collapse to low-rank biases in generative models. In variational inference and flow-matching tasks, we show that this bias can manifest in two different mechanisms: i) where collapsed models are stable fixed points of the gradient flow, or ii) in sequential feature learning, where smaller modes take progressively longer to be learned, producing an effective collapse for a fixed training budget. Together these findings point to a trade-off between feature learning and collapse, suggesting an operator view as a step towards a unified framework for understanding the learning dynamics of dynamical systems-based generative models.
We study the learning of homogenized constitutive laws for inelastic heterogeneous materials, aiming to replace costly concurrent unit-cell simulations with surrogates that generalize across loading histories and microstructures. We focus on the recurrent neural operator framework, motivated by the explicit form of homogenized Kelvin--Voigt linear viscoelasticity. In this framework, internal variables enable a Markovian approximation of loading-history dependence, and neural operators represent the microstructure-dependent constitutive response. For periodic Kelvin--Voigt viscoelasticity, we prove a universal approximation theorem showing that the model can approximate the homogenized constitutive law to any prescribed accuracy, uniformly in time and over a wide set of admissible strain trajectories and microstructural property fields, in two and three dimensions. This comprises a significant extension of the existing, somewhat limited, one-dimensional theory that uses explicit solutions for piecewise constant materials. Instead, in two and three dimensions, we adopt a more versatile approximation based on the spectral representation of the memory kernel that arises from homogenization. This also enables us to give an explicit bound on the internal-variable dimension, sufficient for approximation, exhibiting logarithmic growth in the inverse relative stress error and in material contrast. Numerical experiments on viscoelasticity and crystal plasticity complement the theory, illustrating the roles of internal-variable dimension and sample complexity in learning homogenized responses from unit-cell data. Theoretical results are formally verified in Lean 4.
Reinforcement learning (RL) requires large amounts of interaction data or a reliable simulator, both hard to obtain for nonlinear soft robotic hardware. Recent work shows that a time-scale separation inherent to soft robot dynamics justifies their reduction to low-dimensional invariant manifolds, known as adiabatic spectral submanifolds (aSSMs). We demonstrate that an aSSM-reduced model, a simple polynomial ordinary differential equation learned purely from pre-collected data, can serve as an RL training environment. On a pneumatically actuated soft arm, we train a trajectory-tracking policy in 3 hours on a six-dimensional aSSM-reduced model and deploy it without fine-tuning, whereas reproducing this policy training extrapolates to 3 years of hardware interaction. Our policy tracks dynamic targets with 40% lower error than otherwise identical policies trained on a 4.8-million-parameter recurrent neural network (RNN) or a 1.7-million-parameter transformer, which take 13 and 100 hours to identify, against 5 minutes for our mathematically justified 6,924-parameter model.
A purely data-driven method grounded in the theory of adiabatic spectral submanifolds (aSSMs) was recently used to model and control nonlinear soft robotic simulators. Here, we extend the methodology to learn aSSM-reduced models from data collected from hardware soft robotic platforms. Specifically, we learn a five dimensional (5D) aSSM-reduced nonlinear ordinary differential equation (ODE) that accurately models the dynamics of a soft-helicoid trunk robot. We then use this ODE in a model predictive control scheme to demonstrate real-time precision control across various challenging closed-loop trajectory-tracking tasks, ranging from 2D to 3D trajectories with static and dynamic constraints. We find that our methodology enhances accuracy and precision across all control tasks compared to available data-driven linear methods and model-free reinforcement learning. While we focus here on a soft-helicoid trunk robot, our methodology relies solely on hardware data and is therefore applicable to a broader class of soft robotic systems.
World models learn internal simulators of environment dynamics for prediction, planning, and control, yet learning them from limited, noisy, and partial observations remains challenging. While much progress has focused on pixel-based observations, many physical systems instead provide vector-valued measurements that reveal only part of the underlying state. We introduce K-PWM, a probabilistic world model for prediction and control under partial observation. K-PWM couples Koopman-bilinear latent dynamics with a nonlinear observation decoder and performs probabilistic latent-state inference through the learned model via extended Kalman filtering and smoothing, without a separately trained inference network. Its bilinear structure enables exact propagation of the maintained Gaussian belief through the latent transition and efficient rollouts for uncertainty-aware model predictive path integral (MPPI) control. For training K-PWM, we present a generalized expectation conditional maximization (GECM) procedure with closed-form updates for the generators, noise covariances, and latent prior, complemented by gradient-based decoder learning. A Koopman-consistent initialization selects the latent dimension from data and initializes the action-dependent dynamics. With limited training data, K-PWM achieves lower long-horizon prediction error than the evaluated baselines on partially observed Gymnasium MuJoCo systems, while substantially reducing training time relative to the strongest prediction baseline. On high-dimensional MuJoCo Menagerie robots, it achieves competitive tracking accuracy with low per-step computation times, demonstrating potential for real-time deployment. These results support structured probabilistic dynamics as a data- and compute-efficient approach to world modeling and control under partial observation.
Well-calibrated forecasting for non-linear dynamical systems is essential in science and engineering applications such as fluid dynamics and weather prediction. We present a forecasting method combining probabilistic machine learning with the structured linear dynamics of Koopman mode decomposition. Within this framework, the observation, linear dynamics, and state reconstruction operators are constructed using standard machine learning architectures, producing predictive distributions via the pushforward of injected noise samples. The model is trained using a proper scoring rule, the continuous ranked probability score, to directly target calibrated, accurate forecasts. The approach is validated on two non-linear ordinary differential equation examples and two partial differential equation examples, vortex shedding and cavity flow, using both large and small, noisy datasets. Results demonstrate that known Koopman eigenfunctions and spectra can be identified and used for accurate forecasting. The model can also produce multi-modal predictive distributions, demonstrated for a trajectory initialised on the boundary between two basins of attraction. Uncertainty is captured by the sample ensemble spread, with spread--skill ratios close to one over the training horizon in several experiments, although for extended rollouts and noisy datasets, the ensemble may become under- or over-dispersed. In low-noise experiments, temporal rollout is generally accurate for at least twice the training horizon, and up to twenty times in some cases. Approximation quality is comparable to a deep ensemble approximation over the training horizon, but at a reduced computational cost and with better robustness in identifying Koopman eigenvalues.
Mechanism-AI coupled dynamical systems are increasingly used to study complex biological systems. However, uncertainty associated with noisy, sparse, and partially observed data limits their reliability. This paper presents a systematic mathematical formulation for treating these uncertainties from the perspective of optimal transport and information geometry. We formulate the uncertainty quantification problem as the minimization of an energy functional on a probability density space equipped with a Riemannian metric, with the loss evaluated through a differential equation, and give a unified description of the gradient flows of the energy functional under specific metrics, such as the Wasserstein and Fisher-Rao metrics. Furthermore, we present corresponding numerical methods for solving these gradient flows using abstract adjoint sensitivity analysis. The formulation connects a range of Bayesian inference methods for dynamical systems; numerical experiments compare posterior approximation and predictive uncertainty under a common computational budget. Moreover, it also provides a geometric perspective on dynamical modeling and the study of heterogeneity evolution.
Associative memory models classically describe retrieval as convergence to stored patterns, but the theory of sequential retrieval remains comparatively limited. We develop a two-timescale input-driven Hopfield network in which fast associative retrieval is modulated by a slow feedback variable evolving over a prescribed memory-transition graph. This yields autonomous, graph-constrained transitions while preserving analytical tractability. Using slow-fast and geometric singular perturbation theory, we characterize memory fixation via memory manifolds and identify the geometric regions that govern stability loss and escape from retrieved memories. We further derive reduced slow dynamics and explicit criteria for transition onset, self-sustained retrieval, and collapse, expressed through escape times, gain thresholds, and fixed points. Together, these results provide a tractable framework for analyzing feedback-driven sequential retrieval in Hopfield networks.
We prove that for every open $U \subseteq \mathbb{R}^n$ the spherical projection of every gradient trajectory $\theta(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb{R}$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We extend our conclusions from the Euclidean metric to a general class of ambient metrics including Riemannian metrics and we establish uniform local upper bounds for the lengths of the spherical projections of gradient flow trajectories. We finally apply our findings to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).
We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for Gaussian priors, showing the empirical estimator converges to the Bayes-optimal predictor under sequential asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.
Ample empirical evidence in deep neural network training suggests that a variety of optimizers tend to find nearly global optima. In this article, we adopt the reversed perspective that convergence to an arbitrary point is assumed rather than proven, focusing on the consequences of this assumption. From this viewpoint, in line with recent advances on the edge-of-stability phenomenon, we argue that different optimizers effectively act as eigenvalue filters determined by their hyperparameters. Specifically, the standard gradient descent method inherently avoids the sharpest minima, whereas Sharpness-Aware Minimization (SAM) algorithms go even further by actively favoring wider basins. Inspired by these insights, we propose two novel algorithms that exhibit enhanced eigenvalue filtering, effectively promoting wider minima. Our theoretical analysis leverages a generalized Hadamard--Perron stable manifold theorem and applies to general semialgebraic $C^2$ functions, without requiring additional non-degeneracy conditions or global Lipschitz bound assumptions. We support our conclusions with numerical experiments on feed-forward neural networks.
Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.
In this paper, we present a Koopman autoencoder-based least-squares policy iteration (KAE-LSPI) algorithm for reinforcement learning (RL). The KAE-LSPI algorithm is based on reformulating the so-called least-squares fixed-point approximation method in terms of extended dynamic mode decomposition (EDMD), thereby enabling automatic feature learning via the Koopman autoencoder (KAE) framework. The approach is motivated by the lack of a systematic procedure for selecting features or kernels in linear RL techniques. Unlike previous approaches, no features or kernels need to be fixed a priori in our method. We discuss the theoretical convergence of KAE-LSPI under suitable assumptions, similar to kernel-based least-squares policy iteration (KLSPI). We empirically compare our method with two existing algorithms: least-squares policy iteration (LSPI) and KLSPI. In particular, for the chainwalk problem, the resulting policies are either optimal or nearly optimal for both three approaches. For the cart-pole system, KAE-LSPI outperforms the other two approaches in terms of policy performance.
Discovering continuous symmetries directly from differential-equation data can reveal geometric structure for model reduction and scientific machine learning without symbolic determining equations or a closed-form representation of the governing system. We formulate intrinsic Lie point-symmetry discovery as an operator-theoretic problem on sampled jet-space data. Admissible generators are identified from a sampled determining operator and separated from redundant, solution-dependent, and geometrically trivial directions. The framework combines prolongation, sequential subspace restrictions, and a fixed nonlinear dictionary linear in trainable coefficients. A rank-stabilization preflight test diagnoses systems whose represented intrinsic symmetry algebra does not stabilize under dictionary refinement. On a controlled differential-equation benchmark with an analytically known 11-dimensional represented intrinsic symmetry algebra, the method recovers the complete represented generator set, reproduces its Lie-bracket structure to numerical precision, and approximates a known generator beyond the training domain. Here we show that explicit intrinsic continuous-symmetry generators can be recovered from data obtained from exact or approximate differential systems.
Dooge asked whether hydrology has laws at the catchment scale. We ask whether a low-dimensional system built under physical laws can match the accuracy of neural networks that outperform process-based models on the same inputs. Three laws constrain a lumped catchment model. Conservation makes it a flux-connected storage network, dissipation fixes the sign of every flux and aggregation determines how the unresolved composition enters the fluxes, through effective laws over threshold distributions and a memory of the forcing. These laws admit four kinds of refinement. A parsimonious model under them, CaRT (11 parameters per catchment), is refined in a loop assisted by AI agents. Each refinement addresses an observed deficit through a hypothesis from the literature. The result, CaRT+, has fourteen states, 35 parameters per catchment and no daily neural network. Most refinements follow from aggregation over thresholds or from variables that carry no water but modulate fluxes. On 602 British and 593 US gauged catchments, single CaRT+ models reach median Nash-Sutcliffe efficiencies of 0.863 and 0.765 in held-out years. A stateful LSTM reaches 0.869 and 0.751 and an entity-aware LSTM 0.862 and 0.754, while three-seed LSTM ensembles remain ahead of the CaRT+ ensemble by 0.003-0.012 in median paired NSE. On 505 drier Australian gauged catchments unused in development, single CaRT+ models reach 0.778 against 0.762 and 0.753 for the two LSTMs. The results suggest that catchment-scale laws exist as laws of form, with the constitutive families shared and only their parameters learned.
Koopman autoencoders (KAEs) seek a higher-dimensional latent representation in which nonlinear dynamics evolve linearly. However, many interesting systems have multiple basins of attraction, and both theoretical and empirical work has shown these multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding under standard assumptions. We posit that encoders with a sparsity-inducing objective encouraging few active latent coefficients will provide latent supports as an inspectable basin-modeling principle for Koopman autoencoders. We use these encoders producing sparse latents in training Sparse Koopman Autoencoders (SKAEs) without basin labels or other regime annotations, and treat the learned latent supports as model-produced regime variables after training. Across a range of procedurally generated multibasin systems and chaotic flows, we show that SKAEs have superior forecasting performance compared to dense-latent KAEs. We also perform a mechanistic study that shows latent supports produced by SKAEs are both essential for the quality of the representation and useful for identifying basins on basin-interior states, whereas dense-latent KAEs collapse to an uninformative single support family. These results identify sparse latents and their corresponding supports as label-free, interpretable regime variables for Koopman learning in nonlinear systems with multiple local dynamical laws.
We introduce RaNNDy, a randomized neural network framework for learning transfer operators and their spectral decompositions. By fixing the weights of the hidden layers and training only the output layer, RaNNDy achieves significant reductions in computational costs -- often by two orders of magnitude -- with similar accuracy compared to state-of-the-art deep learning methods. The method is less sensitive to hyperparameter tuning and provides a closed-form solution for the output layer weights that directly represent eigenfunctions of the learned operator. Moreover, it is possible to estimate uncertainties associated with the computed spectral properties via ensemble learning. We demonstrate the efficacy of the proposed approach using diverse benchmark problems, including stochastic dynamical systems and protein folding processes, highlighting the strengths but also weaknesses of RaNNDy.
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.