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J. Nathan Kutz

Publications and source records attributed to J. Nathan Kutz.

At least 19 recordsLinked to original sources
arXiv2026

UQ-SHRED: uncertainty quantification of shallow recurrent decoder networks for sparse sensing via engression

Reconstructing high-dimensional spatiotemporal fields from sparse sensor measurements is critical in a wide range of scientific applications. The SHallow REcurrent Decoder (SHRED) architecture is a recent state-of-the-art architecture that reconstructs high-quality spatial domain from hyper-sparse sensor measurement streams. An important limitation of SHRED is that in complex, data-scarce, high-frequency, or stochastic systems, portions of the spatiotemporal field must be modeled with valid uncertainty estimation. We introduce UQ-SHRED, a distributional learning framework for sparse sensing problems that provides uncertainty quantification through a neural network-based distributional regression called engression. UQ-SHRED models the uncertainty by learning the predictive distribution of the spatial state conditioned on the sensor history. By injecting stochastic noise into sensor inputs and training with an energy score loss, UQ-SHRED produces predictive distributions with minimal computational overhead, requiring only noise injection at the input and resampling through a single architecture without retraining or additional network structures. On complicated synthetic and real-life datasets including turbulent flow, atmospheric dynamics, neuroscience and astrophysics, UQ-SHRED provides a distributional approximation with well-calibrated confidence intervals. We further conduct ablation studies to understand how each model setting affects the quality of the UQ-SHRED performance, and its validity on uncertainty quantification over a set of different experimental setups.

cs.LG↗
arXiv2026

Fault-tolerant shallow recurrent decoders for sparse sensing with missing sensor measurements

Real-world sensing systems are inherently imperfect: sensors may fail, intermittently disconnect, operate with asynchronous sampling rates, and be subject to noise, leading to measurement streams that are irregular, incomplete, and heterogeneous. Nevertheless, sensing strategies for real-time state reconstruction typically assume synchronous and continuously active sensors, a constraint that substantially limits their applicability in operational environments. In this work, we introduce fault-tolerant sparse sensing strategies in the context of shallow recurrent decoders (SHRED) to enable high-dimensional state estimation from sparse, asynchronous, and missing sensor measurements. Specifically, we consider naive imputation strategies, fault-tolerant recurrent temporal encoders, and we introduce four different transformer-based masking mechanisms to make SHRED explicitly aware of sensor availability through the binary data-availability mask. Moreover, we further introduce a multi-step decoding training objective for SHRED, resulting in smoother latent trajectories and more robust predictions. Throughout a synthetic benchmark and three real-world problems in hydrodynamic and ocean modeling, we show that (i) transformer-based masking mechanisms outperform standard imputation techniques and fault-tolerant recurrent architectures, (ii) the multi-step decoding training objective enhances reconstruction fidelity and robustness in the presence of heterogeneous missing-data patterns, while (iii) the computational efficiency of SHRED is preserved, with no additional inference-time overhead.

cs.LG↗
arXiv2026

A Taxonomy of Numerical Differentiation Methods

Differentiation is a cornerstone of computing and data analysis in every discipline of science and engineering. Indeed, most fundamental physics laws are expressed as relationships between derivatives in space and time. However, derivatives are rarely directly measurable and must instead be computed, often from noisy, potentially corrupt data streams. There is a rich and broad literature of computational differentiation algorithms, but many impose extra constraints to work correctly, e.g. periodic boundary conditions, or are compromised in the presence of noise and corruption. It can therefore be challenging to select the method best-suited to any particular problem. Here we review a broad range of numerical methods for calculating derivatives, present important contextual considerations and choice points, compare relative advantages, and provide basic theory for each algorithm in order to assist users with the mathematical underpinnings. This serves as a practical guide to help scientists and engineers match methods to application domains. We also provide an open-source Python package, PyNumDiff, which contains a broad suite of methods for differentiating noisy data.

math.NA↗
arXiv2026

Neural Parameter Estimation of RC Thermal Building Models for Model Predictive Control

Gray-box RC models are widely used to enable energy-efficient model predictive control (MPC) in buildings. However, estimating RC parameters remains difficult, as conventional optimization-based algorithms are prone to local minima, rely heavily on good initial guesses, and incur high computational cost. To address these issues, we propose the Estimator from Scratch, a novel neural parameter estimation approach that embeds the physical equations into a neural network's training process to estimate RC parameters. To further improve estimation accuracy and eliminate dependence on an initial guess, we extend this approach by pretraining the neural network on data from multiple source buildings, the Pretrained Estimator. We benchmark both methods against a genetic-algorithm-based RC estimator and a fully black-box neural network. All methods are evaluated across eight simulated and three real-world buildings for two RC configurations, for both prediction accuracy and closed-loop MPC performance, the latter only for the simulated buildings. The Pretrained Estimator achieves the best prediction performance among all RC-based methods, particularly with little training data, and yields the lowest and least variable MPC costs across buildings and benchmarks. These results position the Pretrained Estimator as a robust, computationally efficient, and initial-guess-free alternative for RC parameter estimation, with potential to extend to other control-oriented dynamical systems.

eess.SY↗
arXiv2026

SIPHy: Sparse identification of port-Hamiltonian systems from noisy data

We propose sparse identification of port-Hamiltonian systems (SIPHy), enabling structure-preserving symbolic regression from noisy trajectory observations. The method applies to port-Hamiltonian systems, which provide a general framework for describing dynamical systems in terms of energy exchange, dissipation and control. Our algorithm can jointly identify the Hamiltonian as well as the dissipation and input matrices. Furthermore, we introduce Hamiltonian flow splines to better approximate derivatives of trajectory data corrupted by noise or with missing time points, a major challenge of system identification for differential equations. This method assembles flows of piecewise polynomial Hamiltonians to produce a smooth, differentiable trajectory necessary for sparse regression. Combining flow splines with SIPHy yields interpretable, physically grounded models from noisy and incomplete trajectory data. Because the Hamiltonian, dissipation, and input matrices are identified as separate components, the resulting models can be simulated under control inputs and dissipation regimes never observed during training, thus improving the robustness and generalization capabilities of model discovery methods.

math.DS↗
arXiv2026

Towards Efficient Parametric State Estimation in Circulating Fuel Reactors with Shallow Recurrent Decoder Networks

The recent developments in data-driven methods have paved the way to new methodologies to provide accurate state reconstruction of engineering systems; nuclear reactors represent particularly challenging applications for this task due to the complexity of the strongly coupled physics involved and the extremely harsh and hostile environments, especially for new technologies such as Generation-IV reactors. Data-driven techniques can combine different sources of information, including computational proxy models and local noisy measurements on the system, to robustly estimate the state. This work leverages the novel Shallow Recurrent Decoder architecture to infer the entire state vector (including neutron fluxes, precursors concentrations, temperature, pressure and velocity) of a reactor from three out-of-core time-series neutron flux measurements alone. In particular, this work extends the standard architecture to treat parametric time-series data, ensuring the possibility of investigating different accidental scenarios and showing the capabilities of this approach to provide an accurate state estimation in various operating conditions. This paper considers as a test case the Molten Salt Fast Reactor (MSFR), a Generation-IV reactor concept, characterised by strong coupling between the neutronics and the thermal hydraulics due to the liquid nature of the fuel. The promising results of this work are further strengthened by the possibility of quantifying the uncertainty associated with the state estimation, due to the considerably low training cost. The accurate reconstruction of every characteristic field in real-time makes this approach suitable for monitoring and control purposes in the framework of a reactor digital twin.

cs.LG↗
arXiv2026

Biophysics-informed deep operator learning for inverse problems with application to electrophysiological source reconstruction

Electrophysiological brain signals are typically acquired through indirect and noisy measurements, providing transformed representations of the underlying neural activity. Source reconstruction---the inverse problem of resolving underlying neural signals from these measurements---is essential for mapping brain function but remains challenging because it is ill-posed and sensitive to noise. Deep learning methods have shown promise across a range of inverse problems, yet many do not explicitly incorporate the biophysical principles governing data generation, limiting data efficiency and adaptation across subjects. Here, we introduce DeepOp-Informed, a biophysics-informed geometric deep operator learning framework that embeds the biophysics of the sensing process into the model through a custom differentiable layer, enabling more efficient learning and improved reconstruction performance. This layer enables the neural network to adapt to subject-specific variations in the physics of signal generation, resulting from differences in brain anatomy and sensor positioning. In realistic magnetoencephalography simulations, DeepOp-Informed generalizes to forward models from held-out subjects, reducing reconstruction error relative to several neural-network and classical baselines. Applied to adolescent auditory-evoked recordings, it produces anatomically plausible reconstructions localized to the auditory cortex. While our application focuses on magnetoencephalography, the framework is general and may be adaptable to other imaging modalities.

stat.ME↗
arXiv2026

Stable Takens' Embedding Theorem for Non-Uniformly-Sampled Linear Systems

Takens' time-delay embedding theorem provides conditions under which delay-coordinate maps, formed using uniformly-sampled time series of trajectories evolving on attractors of dynamical systems, can faithfully represent the dynamics of the original system. Nonlinear systems can be highly sensitive, and Takens' theorem does not provide guarantees about the stability of time-delay embeddings. In the linear setting, statements about the stability of time-delay embeddings are more tractable and have been proven for delay-coordinate maps with evenly-spaced delays. In many experimental applications, however, time series data may be non-uniformly-sampled, especially in systems with multiple timescales or when using event-based rather than time-based sampling techniques. In this paper, we extend the theorems for the stable linear Takens' embeddings to the setting where the delay-coordinate maps involve unevenly-spaced delays. We pose a conjecture about the rank of generalized Vandermonde matrices that capture the temporal structure of time-delay embeddings. We prove that, provided the conjecture holds, existing theorems about stable linear Takens' embeddings readily extend to unevenly-sampled settings, and the quality of the embedding converges to the same asymptotic bounds when using a large number of delays in the delay-coordinate map, a result which is supported by numerical simulations.

math.DS↗
arXiv2026

Mapping surface height dynamics to subsurface flow physics in free-surface turbulent flow using a shallow recurrent decoder

Near-surface turbulent flows beneath a free surface are reconstructed from sparse measurements of the surface height variation, by a neural network algorithm known as the SHallow REcurrent Decoder (SHRED). The reconstruction of turbulent flow fields from limited, partial, or indirect measurements remains a grand challenge in science and engineering. The central goal in such applications is to leverage easy-to-measure proxy variables in order to estimate quantities which have not been, and perhaps cannot in practice be, measured. In the application considered here, the aim is to use a sparse number of surface height point measurements of a flow field, or drone video footage of surface features, in order to infer the turbulent flow field beneath the surface. SHRED is a deep learning architecture that learns a delay-coordinate embedding from a few surface height (point) sensors and maps it, via a shallow decoder trained in a compressed basis, to full subsurface fields, enabling fast, robust training from minimal data. We demonstrate the SHRED sensing architecture on two types of turbulent data from recent studies [Aarnes et al. J. Fluid Mech. 1007, A38 (2025) and Babiker et al. Phys. Rev. Fluids 11, 054802 (2026), respectively]: fully resolved DNS data and PIV laboratory data from a turbulent water tank. SHRED is capable of robustly mapping surface height fluctuations to full-state flow fields up to about one integral length scale deep, with as few as three surface measurements.

physics.flu-dyn↗
arXiv2026

Origins and mitigation of double descent in reduced order modeling

Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements. Depending on the reconstruction algorithm, sensor locations, and measurement noise, the reconstruction risk curves demonstrate a diversity of patterns including a dramatic peak in error known as double descent in Machine Learning literature. Here we explore those scenarios under a unified Data-Noise Averaging theory. Qualitatively, we formulate sufficient criteria for double descent to emerge through a catastrophic amplification of a pathological signal in reconstruction. Quantitatively, we predict the detailed risk curves at a fraction of computational cost, trace reconstruction instability to individual sensors and their combinations, and provide regularization mechanisms to mitigate the instability. We demonstrate results for both static reconstruction of Sea Surface Temperature patterns and time integration of a reduced order model of a PDE.

stat.ML↗
arXiv2026

Multi-Fidelity Learning with Shallow Recurrent Decoders for Multi-Physics Applications

In reactor physics, neutronics and multi-physics phenomena can be modelled at different fidelity levels. High-fidelity models based on the Boltzmann transport equation, multi-group diffusion, or computational fluid dynamics are computationally demanding, whereas simplified models, such as zero-dimensional lumped formulations, can be evaluated efficiently at the cost of neglecting spatial details. The computational intractability of detailed models translates into a scarcity of high-fidelity data and an abundance of low-fidelity data, motivating the development of multi-fidelity (MF) learning strategies able to map between the two. This work extends Shallow Recurrent Decoders (SHRED), a machine learning architecture that reconstructs high-dimensional fields from time-series measurements, to multi-fidelity reduced-order modelling, in which the input trajectories are provided by a low-fidelity model. The resulting MF-SHRED is assessed on three benchmark problems: (i) a two-group point-kinetics-to-diffusion; (ii) a non-linear reaction-advection-diffusion system of six chemical species; and (iii) the coupled neutronics-thermal-hydraulics multi-physics model of the Molten Salt Fast Reactor (MSFR). Across all three cases, MF-SHRED reconstructs the high-fidelity fields with relative errors below a few percent, closely approaching the truncation error of the underlying proper orthogonal decomposition, while reducing the computational cost by three orders of magnitude relative to the corresponding high-fidelity solver. MF-SHRED performs comparably to the original sparse-sensor SHRED formulation. These results support the use of MF-SHRED as a non-intrusive reduced-order modelling strategy for reactor physics and multi-physics applications, particularly for design and safety analysis tasks that must be carried out before a facility is even built.

physics.comp-ph↗
arXiv2026

Real-time optimal control with shallow recurrent decoder networks

Controlling dynamical systems in real-time across multiple scenarios is critical to enabling adaptive control strategies, ensuring stability and efficiency. However, to tailor control actions in response to varying scenarios, traditional optimal control problems typically require several system simulations, which are often computationally demanding due to the high-dimensionality of the underlying spatio-temporal dynamics. In this work, we exploit SHallow REcurrent Decoder networks-based Reduced Order Modeling (SHRED-ROM) to synthesize a real-time closed-loop controller for high-dimensional and parametric dynamics, relying solely on limited state sensor readings. After training the model on a few optimal examples given by an expert demonstrator, SHRED-ROM mimics the expert behavior with effective distributed control actions in new scenarios, alleviating the curse of dimensionality. Moreover, a sensor forecaster is synthesized and used to close the loop at the latent level, thus efficiently mitigating possible sensor failures or delays. The performance of the proposed optimal control strategy is finally assessed on three challenging high-dimensional cases dealing with either parametric density control or fluid flow control.

cs.LG↗
arXiv2026

Data-Driven Discovery of Multiscale Power System Oscillation Governing Equations Using SINDy-SENDAI

Monitoring electromechanical oscillations is crucial for maintaining the stability of modern power systems, particularly in the presence of increasing penetrations of inverter-based resources (IBRs), which introduce new dynamic behaviors. In this work, we propose a hierarchical multiscale framework based on the SINDy-SENDAI algorithm to characterize the transient dynamics captured by wide-area measurements. The proposed deep learning architecture robustly separates low- and high-frequency components embedded in sensor data and incorporates a Sparse Identification of Nonlinear Dynamical Systems (SINDy) module in the latent space to identify parsimonious governing equations. In contrast to conventional deep learning approaches that often produce black-box models with limited interpretability, the proposed framework learns an explicit dynamical representation, enabling physical interpretation, stability assessment, and forecasting of electromechanical oscillations. Given the societal importance of modern power systems, the proposed approach is specifically designed to satisfy key requirements for practical deployment, namely robustness, interpretability, and stable performance under diverse operating conditions. The framework is first validated on the two-area Kundur test system using conventional modal analysis as ground truth and subsequently demonstrated on two real-world datasets: the 2016 Iberian oscillatory event and the 2021 ambient measurements from the southern Italian power grid. The results show that SINDy-SENDAI consistently outperforms the state-of-the-art Hankel-DMD method and that the learned latent dynamics are sufficiently informative to accurately reconstruct and predict the behavior of the full system in the original state space.

eess.SY↗
arXiv2026

A Shallow Recurrent Decoder for Dynamic State Estimation with a Limited Number of PMUs in Power Systems

Dynamic State Estimation (DSE) will play a fundamental role in future power system operation by providing real-time estimates of the system state and enabling enhanced situational awareness. Existing DSE approaches are primarily based on Kalman filter variants or Machine Learning (ML) techniques. However, Kalman-based methods often suffer from high computational complexity, sensitivity to model inaccuracies, and performance degradation under strongly nonlinear operating conditions. Moreover, their effectiveness critically depends on the number and placement of measurements, since suboptimal PMU locations can reduce observability and even render state estimation infeasible. Machine learning approaches alleviate some of these limitations but typically require large amounts of training data and may struggle to generalize. To address these challenges, this paper proposes a SHallow REcurrent Decoder (SHRED) architecture for full-state reconstruction of power systems from sparse measurements. Unlike conventional model-based estimators, the proposed approach does not rely on an accurate physical model and is largely insensitive to PMU placement, making it particularly attractive for practical deployment in existing Wide Area Measurement Systems (WAMS). The method is validated on the IEEE 39-bus system under strongly nonlinear conditions, including short-circuit disturbances. The results demonstrate that SHRED can accurately reconstruct the complete system state using only a limited number of PMU measurements, consistently outperforming a state-of-the-art shallow decoder benchmark in sparse-measurement scenarios. Furthermore, the proposed framework exhibits strong robustness to measurement noise and maintains high reconstruction accuracy even under severe disturbances, highlighting its potential as a scalable and reliable alternative to conventional DSE techniques.

eess.SY↗
arXiv2026

PRONE: Petrov-Galerkin Operator Learning Unifies DMD, SINDy & Koopmanism

Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov Regression Of Nonlinear Evolution (PRONE), a Petrov--Galerkin regression framework based on $ Ψ(\mathbf{X})K \approx Φ(\mathbf{Y}), $ with distinct trial and test dictionaries. In this form, DMD, EDMD, SINDy, Koopman regression, sparse regression, and low-rank regression become variants of one construction: different dictionaries, weights, and constraints. We keep the linear algebra of Koopman learning, but drop the artificial requirement that a finite model map a dictionary into itself. With this asymmetry, eigenmodes are no longer the right objects. Instead, we use singular modes, which identify the observable combinations captured by the data, their projected futures, and the strength of the coupling between the two spaces. We identify the limiting projected operator and prove $L^2$ convergence of the resulting nonlinear predictor. We give examples from chaotic maps, the double gyre, a pitching-airfoil wake, and Lorenz--63, where PRONE outperforms DeepONets, Fourier neural operators, and reservoir computers with considerably fewer parameters. These examples show the same message: lift once, regress once, and let the singular structure reveal statistics, transport, prediction, and dimension.

math.DS↗
arXiv2026

Multiscale Decomposition Reveals Predictable Interannual Variability and Climate Trends in Antarctic Sea Ice Loss

Antarctic sea ice has undergone unprecedented changes in recent years, raising questions about how this key geophysical system is responding to climate change. Decades of slow expansion were replaced by a precipitous decline in 2014-2017, a subsequent apparent recovery, and a renewed collapse from 2022 to the present. We diagnosed sea ice concentration (SIC) from satellite observations with a hierarchical decomposition method based on Dynamic Mode Decomposition (DMD) that finds coherent spatiotemporal modes. We find that the 2014-2017 decline and apparent recovery are the result of interacting interannual modes and that a climate change signal emerges in 2012, which becomes unambiguous by 2022 when it dominates over interannual variability. These rapid changes underscore the need for seasonal-to-annual forecasts of SIC. However, existing forecasts are subject to limited prediction horizons combined with high computational costs. Our predictive DMD model (IceDMD) is regularised to prioritize the stationary spatiotemporal modes found by the decomposition. The predictive model can forecast SIC anomalies in 2023-2024 up to two years in advance, outperforming all existing approaches with the additional benefits of physical interpretability and extremely cheap computational cost. Finally, this framework for regularising predictive DMD models can be generalized to a range of multi-scale systems.

physics.ao-ph↗
arXiv2026

Linear Recurrent Neural Networks as Time-Delay Embeddings

Sequence models, and particularly Linear Recurrent Neural Networks (LRNNs) of the form $\mathbf{h}_{k+1} = \mathbf{W} \mathbf{h}_{k} + \mathbf{y}_k + \mathbf{b}$, are widely applicable in time-series analysis for dynamical systems, yet, as black-box algorithms, much is unknown about why they perform well. In this work, we leverage Takens' embedding theorem, which provides conditions under which partially observed time series organized into delay-coordinate vectors can faithfully represent the original system's dynamics, as a theoretical framework for explaining how and why sequence models preserve and reconstruct dynamical systems. For LRNNs, concatenating output states into delay-coordinate vectors gives rise to a ``delay" matrix $\mathbb{M}_{n,m}\in \mathbb{C}^{(nm) \times (n+1)m}$: a block matrix consisting of identity matrices $\mathbf{I} \in \mathbb{R}^{m \times m}$ repeated $n$ times along the main diagonal and weight matrices $\mathbf{W} \in \mathbb{C}^{m \times m}$ featured $n$ times along the super-diagonal. $\mathbb{M}_{n,m}$ relates the delay-coordinates of the input time series to those of the LRNN output states, and, for $\mathbb{M}_{n,m}$ to be an embedding, it must be full row-rank. We provide explicit conditions for $\mathbb{M}_{n,m}$ to be full row-rank and prove the condition number of $\mathbb{M}_{n,m}$ and determinant of $\mathbb{M}_{n,m} \mathbb{M}_{n,m}^*$--measures of embedding stability--are bounded independent of $n$, at least for certain ranges of $\mathbf{W}$'s singular values: namely, when $σ_{\max}(\mathbf{W}) \le 1$. This result explains why the spectrum of $\mathbf{W}$ for trained LRNNs tends to converge to within the unit circle.

math.DS↗
arXiv2026

CTF4Nuclear: Common Task Framework for Nuclear Fission and Fusion Models

The demand for clean energy is ever increasing, with new nuclear technologies presenting a complementary solution to renewable energies. However, designing and operating these systems is exceptionally difficult, given the complexity of the physical phenomena that interact to form the system dynamics. While high-fidelity simulations help to understand the non-linear, multi-physics interactions within a reactor, they are computationally expensive and rarely suitable for real-time applications. Furthermore, model-based approaches are inherently sensitive to simplifying assumptions required to derive their governing equations and parameters, leading to inevitable discrepancies with real-world measurements. In contrast, Machine Learning (ML) methods have the potential to generate reliable surrogate models which may be able to quickly predict the system's behaviour. However, the number of data-driven methods that can potentially be used for this task is large and diverse. In a safety-critical setting such as nuclear engineering, a fair comparison of different ML methods, and a clear understanding of their advantages and limitations, is of paramount importance. To address this, we introduce a Common Task Framework (CTF) for ML in nuclear engineering, building upon previous efforts in dynamical systems and seismology. This CTF considers a curated set of datasets from different nuclear and nuclear-adjacent systems. The CTF evaluates the performance of a method on 12 established metrics, alongside a new paradigm focused on system monitoring from sparse measurements only. We illustrate the framework by benchmarking standard ML baselines against these datasets, revealing current method limitations. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigour and reproducibility in scientific ML for the nuclear industry.

cs.LG↗