The discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace, and obtains a trajectory-informed spectral preconditioner, in which the subspace is unchanged but the basis is reweighted and realigned toward the modes visited by the dynamics.
Abstract
Reduced-order models compress high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they lose accuracy when online dynamics drift beyond the training data. Adaptive methods address this by updating the subspace online with external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. By streaming the model's own predictions through an incremental singular-value decomposition with forgetting, we obtain a trajectory-informed spectral preconditioner, in which the subspace is unchanged but the basis is reweighted and realigned toward the modes visited by the dynamics. This enables the model to absorb future out-of-span corrections more effectively. We expose aspects of this mechanism on a three-dimensional spiral and confirm it on viscous Burgers and Fisher-KPP dynamics. We also discuss how in-span learning can be viewed as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.
This work forms AdamW as a finite-horizon input--state--output (ISO) system whose state contains the model parameters and first- and second-moment estimates, and derives an exact multistep error decomposition and establishes first-order finite-horizon accuracy under local smoothness and controlled activation switching.
Modern deep networks are trained through long update trajectories, yet their temporal organization remains less systematically characterized than architectures, losses, or optimizers. We study short-horizon predictability as a measure of temporal redundancy: where, when, and under which training conditions recent updates contain information about near-future parameter motion. We combine three complementary probe families, displacement-direction, subspace-residual, and predictor-based probes, with convention-aware, null-calibrated group-level readouts, and apply them to multi-pass vision training on CIFAR and public Pythia pretraining checkpoints. Across both regimes, vector-like tensors such as normalization parameters and biases (auxiliary parameters) exhibit simpler short-horizon dynamics than matrix-like feature-transforming weights (bulk parameters), whose predictable behavior concentrates in localized, time-varying pockets. Agreement within and across probe families, and with independent trajectory diagnostics, indicates that these measurements capture intrinsic trajectory structure, while probe differences distinguish complementary forms of temporal organization. Controlled CIFAR comparisons further show that architecture and training recipe systematically modulate the measured structure. A Pythia-70M case study further exposes a sequence of role-, depth-, and scale-dependent events, including bulk ESA falling below the random sign-agreement level and the emergence and redistribution of predictable qkv pockets across layers. These results position short-horizon predictability as a retrospective, parameter-resolved diagnostic of training dynamics.
Learning a compact model of the world from interaction data is central to sample-efficient deep reinforcement learning. Spectral representation methods have become the leading paradigm for representation learning in continuous control by taking a matrix view of the transition kernel, with state-action pairs on one side and next states on the other, and learning a low-rank factorization through self-supervised contrastive objectives. We take this view one step further. The transition kernel is naturally a three-mode tensor over states, actions, and next states, and a CP decomposition gives one feature map per mode. We propose FaStR, which fits this decomposition with a noise contrastive objective, producing separate state, action, and next-state encoders that together form a single spectral representation. The factored form yields a smaller hypothesis class, and the sample size needed for representation learning shrinks by a factor that scales with the smaller of the state and action dimensions. Empirically, FaStR delivers its largest gains on high-dimensional locomotion tasks whose dynamics align with the factored structure, and the learned state encoder transfers intact across actuator shift while only the action encoder is retrained.
Intelligent systems should not only solve tasks but also adapt under real-world constraints. Autonomous adaptation via self-supervised learning, sequential adaptation via online learning, and memory-efficient implementation via perturbation-based learning are important requirements for such systems. However, these requirements are generally in tension for high-dimensional systems, because perturbation-based learning suffers from variance that grows with the dimension of the perturbed variables. In this study, we focus on echo state networks (ESNs), where this tension naturally arises in large reservoirs. We propose a perturbation-based learning rule for online self-supervised learning in ESNs. The proposed rule is derived from an orthogonal decomposition of the self-supervised learning cost, which separates an input-dependent component from a redundant component determined by the fixed ESN parameters. By perturbing only the input-dependent component, the effective perturbation dimension is reduced from the reservoir dimension to the input dimension. Thus, the proposed method preserves self-supervised adaptation, online learning, and scalar-feedback perturbation learning, while avoiding reservoir-size-dependent variance growth. This suggests a design principle for scalable and hardware-compatible learning: online learning should be restricted to the dynamically necessary low-dimensional component of the objective.
A Fourier-enhanced operator autoencoder for decoder-free reconstruction and latent learning of dynamical systems and achieves accuracy comparable to or better than classical AE-based reduced-order models while providing a more efficient latent-to-field reconstruction path.
Xuandong Lu, Yongming Liu· Machine Learning for Computa...· 0 citations