Coefficients

Prompt learning adapts CLIP to downstream recognition by replacing hand-written templates with learned continuous context vectors, which in Context Optimization (CoOp) form a dense prompt matrix $\mathbf{P}\in\mathbb{R}^{m\times d}$ trained from only a few examples per class. We study whether this matrix is over-parameterized by factorizing it as $\mathbf{P}=\mathbf{B}\mathbf{A}$, which cuts the trainable prompt parameters from $md$ to $r(m+d)$, and to $rd$ once the token-side factor $\mathbf{B}$ is fixed. Across seven few-shot benchmarks and two CLIP backbones, low-rank prompts match or improve dense CoOp at far fewer parameters, with the clearest gains on low-shot base-to-new generalization. We then find that the token-side factor need not be learned at all: fixing $\mathbf{B}$ to a Gaussian, orthogonal, SVD-derived, or even random basis and training only the embedding-side factor $\mathbf{A}$ stays on par with the fully trainable factorization, and a source-trained $\mathbf{B}$ offers no advantage over a random one. A prompt-factor asymmetry and a local update-space dimension gap show why fixing $\mathbf{B}$ is far less restrictive than fixing $\mathbf{A}$, and a smoothness-only guarantee certifies that optimizing $\mathbf{A}$ over a fixed $\mathbf{B}$ converges. In the CLIP prompt setting, the embedding-side coefficients carry the adaptation while the token basis can simply be fixed.

Data-driven discovery of governing equations from spatiotemporal data offers a promising route to obtaining coarse-grained descriptions of complex dynamical systems. Here, we investigate the performance of PDE-SINDy for discovering phase-ordering dynamics using the Allen--Cahn equation as a benchmark and the Ising model with Glauber spin-flip dynamics as a microscopic system. We systematically analyze the effects of data availability, size of the candidate library, and noise on the efficiency of the equation discovery. We find that stability-selection PDE-SINDy can robustly identify the relevant terms in the governing dynamics even under limited or noisy data, while the recovered coefficient values are substantially more sensitive to these factors. We further show that enlarging the candidate library can strongly affect both term identification and coefficient recovery. Incorporating library bagging with stability selection reduces this sensitivity and improves the efficiency of equation discovery. For the Glauber spin flip Ising model dynamics, the resulting coarse-grained equation reproduces the characteristic phase-separation and coarsening dynamics of the underlying microscopic system. Overall, our results demonstrate the potential of PDE-SINDy for phase-ordering systems while highlighting the importance of carefully assessing the factors that influence the efficiency of equation discovery.

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

We investigate the extent to which totally positive matrices can be distinguished through the highest-order coefficients of their characteristic polynomials. To identify the most informative coefficients, we also employed neural-network classifiers together with feature-attribution methods. Using datasets built from several structured totally positive families, including products of positive bidiagonal matrices, Vandermonde matrices, and Cauchy matrices, we find that the coefficients (a_{n-1}, a_{n-2}, a_{n-3}) already contain strong discriminatory information for separating totally positive from non-totally positive matrices in dimensions 5, 10, and 30. The resulting separation is markedly nonlinear and admits a natural geometric description in the corresponding three-dimensional coefficient space by means of Mahalanobis ellipsoids. These ellipsoids enclose the totally positive samples while excluding most non-totally positive ones. Moreover, different structured totally positive families exhibit distinct ellipsoidal signatures, and the separation between these signatures increases with the dimension. These observations lead us to formulate a conjecture on the geometric separation of structured totally positive families in the space determined by the three highest-order characteristic coefficients.

The paper presents a new parameter-efficient adaptation method called ChebyMA (Chebyshev Manifold Adaptation). ChebyMA adopts weight matrices through a multi-surface superposition of Chebyshev polynomial bases evaluated on learnable coordinates and combined via trainable coefficient matrices, replacing standard linear projections with highly expressive continuous function approximation. Theoretically, we establish an Approximation Expressivity Theorem, proving from the perspective of function approximation theory that single-manifold ChebyMA guarantees convergence in Frobenius norm error of reconstruction. Besides, drawing on Kolmogorov $n$-width intuition, we demonstrate the expressive advantages of multi-manifold superposition ($S > 1$) in decoupling high-dimensional complex features. Experimental results on Computer Vision CIFAR datasets(CIFAR-10, CIFAR-100)\cite{CIFAR} and Natural Language Processing (AG News, SST-2) datasets demonstrate that ChebyMA consistently achieves a superior parameter-accuracy Pareto front compared to standard full-parameter fine-tuning, LoRA\cite{LoRA}, TLoRA\cite{TLoRA}, and StelLA\cite{StelLA}. ChebyMA significantly outperforms other tested methods in tested datasets, validating its solid theoretical foundation for generality with purely vectorized computations.

Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differential operators and dynamical systems. For a single input-response pair, recovery is possible exactly when the evaluated dictionary terms are linearly independent. We prove a sharp zero-one law: either no input uniquely determines the coefficients, or almost every random input sampled from a nondegenerate Gaussian measure does. This dichotomy reduces one-shot system identification to a question about degenerate inputs and provides an a posteriori certificate for any recovered model. Numerical examples recover dynamical systems, nonlinear partial differential equations, and structured matrix families from single trajectory data, while also detecting when an extra probe is necessary.

Humans recognize movements effortlessly, even from noisy and complex visual input. But what information in the stimulus allows humans to rapidly classify movements? No framework has systematically compared different strategies of movement analysis to address this question. Here, we used videos of 16 daily activities from the MoVi dataset and compared three strategies: Temporal Movement Primitives (TMPs), which decompose movements into weighted sums of temporally smooth basis functions; Legendre polynomial coefficients, which project joint-coordinate trajectories onto an orthogonal polynomial basis; and Autoencoder latent embeddings. Legendre coefficients and TMPs achieved the highest classifier accuracy, followed by autoencoders. We found two discriminative features for movement classification. The most informative is the general posture of the body, the average spatial configuration that distinguishes one activity from another. Additionally, we identified 9 critical joints that are most predictive for movement classification. Interestingly, good classification accuracy did not automatically lead to good movement generation: when we reconstructed movements for each activity, TMPs preserved the temporal dynamics and produced perceptually natural motion, whereas reconstructions from Legendre coefficients retained only the average posture and appeared frozen. These results reveal a dissociation in how movement information is organized: the static configuration of the body suffices to classify what activity is performed, but the temporal dynamics of movement are required to reconstruct how it unfolds. This distinction clarifies which features the visual system may rely upon for rapid action recognition, and suggests that postural features could enable efficient movement screening in clinical applications, while dynamic information remain essential wherever movement generation is the goal.

Neural scaling laws describe how pre-training loss decays as power laws with training time, model size, and compute. This position paper argues that the exponents of these power laws are fixed by generic mechanisms: a one-third time scaling due to the strong nonlinearity of Softmax, an inverse width scaling due to representational superposition, and an inverse depth scaling due to ensemble averaging of Transformer layers. These mechanisms are robust to a wide range of data structures and architectural details, placing current large language models in a universality class with fixed exponents. The coefficients, however, are expected to be sensitive to data and architecture details, and directly determine practical quantities such as the optimal model shape and the compute-optimal frontier. We therefore argue that understanding the coefficients is the key to near-term performance improvements, and that a closer examination of the current universality class may reveal pathways to better universality classes.

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

Diffusion models are known to exploit unknown low-dimensional structure to accelerate sampling. However, existing convergence theory under low-dimensional data structure has largely focused on update rules with narrowly prescribed coefficient choices. This raises a fundamental question: is adaptation to low-dimensional structure sensitive to the precise choice of update coefficients? In this paper, we show that such adaptation is a robust property of diffusion models. For a broad class of update coefficients, we prove that $\widetilde{O}(k/\varepsilon)$ iterations suffice to generate an $\varepsilon$-accurate sample in total variation (TV) distance, independently of the ambient dimension. Our framework substantially broadens the class of diffusion samplers known to enjoy low dimensional adaptation and applies to several commonly used methods in practice. These results provide a theoretical justification for the empirical effectiveness of diffusion samplers across different coefficient choices when applied to structured, high-dimensional data.

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

Learning functional relationships from noisy data is a central problem in scientific inference. Spectral methods approximate unknown functions by expanding them in a basis and estimating the corresponding coefficients from data, but the stability of these coefficients under noise remains poorly understood. Here we study supervised regression with additive label noise using sparse spectral representations across multiple bases and dimensions. We show that noise induces a predictable drift in the learned coefficient vector whose magnitude depends on the effective number of active spectral modes. After whitening the empirical feature geometry, we derive a closed-form expression for the overlap between noisy and noiseless coefficient vectors, revealing a universal degradation curve governed by a single intrinsic noise scale. Numerical experiments across Fourier, Legendre, Bessel, and Haar bases confirm the theoretical prediction. The results demonstrate that spectral learning exhibits a fundamental noise threshold beyond which coefficient estimates become unstable, placing intrinsic limits on recovering functional structure from noisy data.

Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when $R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.

Causal discovery with instantaneous effects in multivariate time series is challenging, as the instantaneous structure must be acyclic. Prior methods enforce this by either separating instantaneous and lagged estimation into multi-stage pipelines or imposing algebraic acyclicity constraints via complex augmented Lagrangian optimization, both of which incur high computational cost. In this work, we propose a different approach: we learn a differentiable permutation of variables using the Gumbel--Sinkhorn operator and triangularize the instantaneous coefficient matrix of a Structural Vector Autoregressive (SVAR) model in the learned order. This converts acyclicity from a hard constraint into a parameterization and keeps it valid throughout optimization. In doing so, our method enables unified, continuous optimization with gradient-based learning, leading to improved efficiency in time--series causal discovery. Across three real-world benchmarks, our method achieves the best overall performance compared with 12 baselines in both discovery accuracy and efficiency. On the large-scale benchmark, it further demonstrates strong scalability, achieving more than a 6x speedup over competing methods.

Prediction tasks over individual futures, which are inherently noisy, often admit multiple similarly accurate models. When these models produce different predictions for the same individual, they raise concerns of arbitrariness in decision-making. How severe can this arbitrariness be, in theory and in practice? How can it be resolved to support high-stakes risk assessment? We address these questions through a study of a machine learning-based decision support system for recidivism risk assessment that has been in use for over 15 years. By translating complex legal rules into an algorithm for labeling post release outcomes (recidivist or non-recidivist), we first construct a dataset of thousands of inmate releases. Using this dataset, we learn interpretable models that improve predictive performance, reduce error-rate disparities between groups, and ensure that rehabilitative progress lowers risk scores. Next, we study predictive multiplicity, by first deriving a tight lower bound on the expected predictive agreement of any finite set of models over a dataset, and then by evaluating the extent to which structural diversity (e.g., different model coefficients) within this set translates to predictive multiplicity (i.e., different predictions for the same individual). Our experiments indicate that the existence of many similarly accurate models with comparable error-rate disparities does not necessarily translate into severe predictive multiplicity. Empirically, similarly performant models can exhibit substantially higher predictive agreement than worst-case theoretical guarantees suggest. We find that a simple policy that assigns each inmate the lowest risk among these models is effective for addressing predictive arbitrariness.

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.

Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on complex-valued product-unit networks, in which each unit represents a complex monomial and the network output is a sparse linear combination of such monomials. In contrast to established library-based methods such as SINDy, our approach does not require a predefined set of candidate functions: the relevant monomials, including those with fractional or negative exponents, are learned directly from data. Across four chaotic benchmark systems (Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63), we recover the exact governing equations in 90% of trials for the first three systems, and in 70-90% of trials for the fractional case, using at least 3000 training points. Applied to real-world human-gait accelerometer signals, the model produced stable trajectories with bounded prediction errors, corresponding to an RMSE of approximately 12-14% of the signal amplitude range over a test horizon three times longer than the training interval, demonstrating its potential for high-dimensional systems in which analytic equations are unavailable.

We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net (IV-Net), realizes a mapping from the input coefficients and righthand side to the corresponding solution field. The architecture of IV-Net is informed by, and closely resembles, a V-cycle multigrid solver. The IV-Net model is parameterized via convolutional layers defined in the physical domain. For coercive problems with highly heterogeneous coefficients, the proposed network exhibits superior performance relative to a proper orthogonal decomposition (POD) approach and several existing neural operator architectures. For low-frequency oscillatory Helmholtz problems with smooth coefficients, its performance is similar to that of a Fourier neural operator. We analyze the approximation error and convergence behavior of IV-Net, its data efficiency, and its dependence on the underlying discretization mesh. Furthermore, we demonstrate the practical effectiveness of the architecture through a series of numerical experiments, including applications to uncertainty quantification, inverse problems, and prediction of quantities of interest.

Local interaction laws governing multi-agent systems can be difficult to recover from trajectory data, even when the dynamics are observed faithfully. In systems governed by a nonlinear sheaf Laplacian -- a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels -- the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces. Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node: distinct interaction laws that agree at the node level are indistinguishable from trajectory data alone. We show that the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes. When the obstruction is nontrivial, we show that recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite. Experiments validate the theory and illustrate that accurate trajectory reproduction need not certify recovery of the underlying interaction law.

Learning tractable linear representations of nonlinear dynamical systems via Koopman operator theory is often hindered by dictionary selection, temporal memory encoding, and numerical ill-conditioning. Inspired by Reservoir Computing (RC) paradigm, this paper introduces the RC-Koopman framework, which interprets reservoir as a stateful, finite-dimensional Koopman dictionary whose temporal depth is explicitly controlled by its spectral radius. We show that the Echo State Property (ESP) guarantees well-posedness and favorable numerical conditioning of the lifted Koopman approximation. A correlation-based spectral radius selection algorithm aligns reservoir memory with dominant system timescales. Analysis reveals how the finite memory of the reservoir determines which Koopman eigenfunctions remain observable from the lifted features. Evaluation on synthetic benchmarks demonstrates that RC-Koopman achieves a favorable balance between reconstruction accuracy of the underlying nonlinear dynamics and dynamical stability, compared to Extended Dynamic Mode Decomposition (EDMD) and Hankel-based lifting approaches. Code available at: https://github.com/NEAR-the-future/RC-Koopman.git

In uncertainty quantification, evaluating sensitivity measures under specific conditions (i.e., conditional Sobol' indices) is essential for systems with parameterized responses, such as spatial fields or varying operating conditions. Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space. This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step. Numerical benchmarks indicate that the proposed method ensures physical coherence and offers superior numerical robustness and computational efficiency compared to conventional point-wise approaches.

We present AC-SINDy, a compositional extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that replaces explicit feature libraries with a structured representation based on arithmetic circuits. Rather than enumerating candidate basis functions, the proposed approach constructs nonlinear features through compositions of linear functions and multiplicative interactions, yielding a compact and scalable parameterization and enabling sparsity to be enforced directly over the computational graph. We also introduce a formulation that separates state estimation from dynamics identification by combining latent state inference with shared dynamics and multi-step supervision, improving robustness to noise while preserving interpretability. Experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy.

Nonlinear machine-learning models are increasingly used to discover causal relationships in time-series data, yet the interpretation of their outputs remains poorly understood. In particular, causal scores produced by regularized neural autoregressive models are often treated as analogues of regression coefficients, leading to misleading claims of statistical significance. In this paper, we argue that causal relevance in nonlinear time-series models should be evaluated through forecast necessity rather than coefficient magnitude, and we present a practical evaluation procedure for doing so. We present an interpretable evaluation framework based on systematic edge ablation and forecast comparison, which tests whether a candidate causal relationship is required for accurate prediction. Using Neural Additive Vector Autoregression as a case study model, we apply this framework to a real-world case study of democratic development, modeled as a multivariate time series of panel data - democracy indicators across 139 countries. We show that relationships with similar causal scores can differ dramatically in their predictive necessity due to redundancy, temporal persistence, and regime-specific effects. Our results demonstrate how forecast-necessity testing supports more reliable causal reasoning in applied AI systems and provides practical guidance for interpreting nonlinear time-series models in high-stakes domains.

Data-driven discovery of governing equations has advanced significantly in recent years; however, existing methods often struggle in multiscale systems where dynamically significant terms may have small coefficients. Therefore, we propose Balance-Guided SINDy (BG-SINDy) inspired by the principle of dominant balance, which reformulates $\ell_0$-constrained sparse regression as a term-level $\ell_{2,0}$-regularized problem and solves it using a progressive pruning strategy. Terms are ranked according to their relative contributions to the governing equation balance rather than their absolute coefficient magnitudes. Based on this criterion, BG-SINDy alternates between least-squares regression and elimination of negligible terms, thereby preserving dynamically significant terms even when their coefficients are small. Numerical experiments on the Korteweg--de Vries equation with a small dispersion coefficient, a modified Burgers equation with vanishing hyperviscosity, a modified Kuramoto--Sivashinsky equation with multiple small-coefficient terms, and a two-dimensional reaction--diffusion system demonstrate the validity of BG-SINDy in discovering small-coefficient terms. The proposed method thus provides an efficient approach for discovering governing equations that contain small-coefficient terms.

Quantum feature maps offer expressive embeddings for classical learning tasks, and augmenting sparse identification of nonlinear dynamics (SINDy) with such features is a natural but unexplored direction. We introduce \textbf{Q-SINDy}, a quantum-kernel-augmented SINDy framework, and identify a specific failure mode that arises: \emph{coefficient cannibalization}, in which quantum features absorb coefficient mass that rightfully belongs to the polynomial basis, corrupting equation recovery. We derive the exact cannibalization-bias formula $Δξ_P = (P^\top P)^{-1}P^\top Q\,\hatξ_Q$ and prove that orthogonalizing quantum features against the polynomial column space at fit time eliminates this bias exactly. The claim is verified numerically to machine precision ($<10^{-12}$) on multiple systems. Empirically, across six canonical dynamical systems (Duffing, Van der Pol, Lorenz, Lotka-Volterra, cubic oscillator, Rössler) and three quantum feature map architectures (ZZ-angle encoding, IQP, data re-uploading), orthogonalized Q-SINDy consistently matches vanilla SINDy's structural recovery while uncorrected augmentation degrades true-positive rates by up to 100%. A refined dynamics-aware diagnostic, $R^2_Q$ for $\dot X$, predicts cannibalization severity with statistical significance (Pearson $r=0.70$, $p=0.023$). An RBF classical-kernel control across 20 hyperparameter configurations fails more severely than any quantum variant, ruling out feature count as the cause. Orthogonalization remains robust under depolarizing hardware noise up to 2% per gate, and the framework extends without modification to Burgers' equation.

Classical representation systems such as Fourier series, wavelets, and fixed dictionaries provide analytically tractable basis expansions, but they are not intrinsically adapted to the empirical structure of modern high-dimensional data. Neural networks overcome this limitation by learning features from data, yet they do so through layered nonlinear parameterizations that often sacrifice interpretability, explicit control over basis structure, and mathematical transparency. In this manuscript we develop a non-neural alternative that learns basis functions directly from data through variational optimization. The proposed framework, termed Data Driven Variational Basis Learning (DVBL), treats basis atoms as primary optimization variables and learns them jointly with sample-specific coefficients and, when appropriate, a latent linear evolution operator. This yields a data-adaptive basis expansion that remains explicit, interpretable, and amenable to rigorous analysis. We formulate the model, establish existence of minimizers, prove blockwise descent properties for an alternating minimization algorithm, give conditions for coefficient recovery and basis identifiability, and show how manifold and dynamical regularization can be integrated without invoking neural architectures. We also discuss the conceptual novelty of the framework relative to classical dictionary learning, spectral methods, Koopman operator methods, and deep representation learning.

Classical sparse recovery treats all nonzero entries equally, though numerical noise often creates long tails of negligible coefficients. This paper develops an entropy-based notion of effective sparsity to measure the coefficients carrying significant mass. The central quantity, the effective number of nonzeros (ENZ), is obtained by exponentiating the Shannon entropy of the normalized magnitude distribution. We show that ENZ decomposes exactly into the support cardinality multiplied by a distributional efficiency factor, thereby making precise its relation to the $\ell_0$ count and explaining how it discounts uninformative coefficients. Furthermore, the Shannon ENZ is embedded into a parallel Rényi family that recovers several scale-invariant sparsity measures, including the $\ell_1/\ell_2$ ratio, as special cases. We then prove a stability result under a restricted isometry condition, establishing an explicit bound that depends on the tail energy, measurement perturbation, and restricted isometry constant. For computation, a separable unnormalized entropy surrogate is introduced to avoid global coupling. Numerical experiments on sparse signal recovery and gradient-domain image denoising demonstrate that the resulting regularizer is robust, computationally efficient, and competitive with standard sparsity penalties.

This paper presents a novel Koopman composition operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental equivalences between different model representations, we are able to close the gap of control system operator learning and infinite-dimensional regression, enabling various empirical estimators and the connection to the well-understood learning theory in RKHSs under one unified framework. Consequently, our proposed framework allows for arbitrarily accurate finite-rank approximations in infinite-dimensional spaces and leads to finite-dimensional predictors without a priori restrictions to a finite span of functions or inputs. To enable applications to high-dimensional control systems, we improve the scalability of our proposed control Koopman operator estimates by utilizing sketching techniques. Numerical experiments demonstrate superior prediction accuracy compared to bilinear EDMD, especially in high dimensions. Finally, we show that our learned models are readily interfaced with linear-parameter-varying techniques for model predictive control.