Reduced-Order Modeling

Many dynamical systems can be described in terms of structured flows combining source/sink behavior, cyclic dynamics, and topology-constrained transport. These features arise across a wide range of physical, engineered, and data-driven systems. The objective of this work is to establish a unified perspective on such systems, to identify modeling approaches that balance expressivity, interpretability, computational complexity, and data requirements, and to investigate how highly expressive models can be used to uncover the dominant mechanisms underlying observed dynamics. Starting from the Helmholtz-Hodge decomposition of continuous vector fields, we review the recently proposed Graph Vector Field (GVF) framework and its discrete representation on simplicial complexes. We then introduce a hierarchy of alternative approaches, including parametric conditional models, linear graph dynamical systems, and reduced Hodge representations. Finally, we propose a verification and validation methodology based on benchmark datasets from well-understood physical systems and on systematic model-reduction and ablation studies. The resulting family of structured-flow models within a common framework, ranging from low-dimensional parametric representations to full GVF formulations, supports a diagnostic methodology in which gradient, curl, harmonic, and topological contributions are systematically assessed through ablation studies. This process enables the identification of dominant mechanisms underlying the observed dynamics and guides the construction of simplified models tailored to the available data and operational constraints. By separating structural verification, behavioral verification, and domain-specific validation, the proposed approach provides a foundation for scalable and interpretable analysis of complex dynamical systems across multiple application domains.

pyforce is a Python package implementing Data-Driven Reduced Order Modelling techniques for applications to multi-physics problems, mainly set in the Nuclear Engineering world. The package is part of the ROSE (Reduced Order modelling with data-driven techniques for multi-phySics problEms): mathematical algorithms aimed at reducing the complexity of multi-physics models (for nuclear reactors applications), at searching for optimal sensor positions and at integrating real measures to improve the knowledge on the physical systems. With respect to the previous original implementation based on dolfinx package (v0.6.0), version 1.0.0 of pyforce has been completely re-written using pyvista as backend for mesh importing, computing integrals, and visualisation of results; in addition, functions are stored as numpy arrays, improving the ease of use of the package. This choice allows to use pyforce with any software solver able to export results in VTK format.

Physics-informed operator learning is an attractive candidate for surrogate modeling of microstructures, especially in multiscale finite-element simulations. Its practical use, however, is often limited by the high cost of loss evaluation. We address this bottleneck by combining the Equilibrium Neural Operator (EquiNO) with the QR-based discrete empirical interpolation method (Q-DEIM). EquiNO learns only the modal coefficients of reduced displacement-fluctuation and first Piola-Kirchhoff stress representations built from periodic and divergence-free bases, thereby enforcing periodicity and mechanical equilibrium by construction. Q-DEIM then identifies a small set of spatial points through a column-pivoted QR factorization of the stress basis and restricts constitutive evaluations during training to these points alone. This makes full-batch second-order optimization practical for three-dimensional representative volume elements (RVEs). Homogenized first Piola-Kirchhoff stresses are recovered directly from the offline-averaged reduced stress modes, without the need to reconstruct the full stress field at inference time. We validate the framework on two three-dimensional finite-strain hyperelastic RVEs. Q-DEIM reduces the per-step training cost by roughly three orders of magnitude relative to full-field loss evaluation, while reduced homogenization achieves speed-up factors of order $10^3$ to $10^4$ over direct full-field computations. Despite relying on only a small number of offline snapshot loading paths for basis construction, the method accurately interpolates and extrapolates both microscopic stress fields and homogenized stresses, with prediction quality improving systematically as more snapshots are added.

Physics-based digital twins aim to predict the dynamics of real-world objects under interaction, enabling real-to-sim-to-real applications in robotics. Current approaches reconstruct such twins as explicit physical models (such as spring-mass systems) to predict the dynamics, but the resulting models often inherit the resolution of the visual reconstruction rather than being reduced to the physical complexity required to reproduce task-relevant dynamics. This mismatch introduces redundant topology, making repeated forward-dynamics rollouts unnecessarily expensive. To address this challenge, we present PhySPRING, an fully differentiable GNN-based method to reduce complexity in spring--mass digital twins. PhySPRING jointly learns a hierarchy of coarsened graph topologies and their mechanical parameters from observations. At each reduction level, PhySPRING merges nodes with similar learned dynamic responses to optimize the topology, while maintaining every reduced layer as an explicit spring--mass system. On the PhysTwin benchmark, PhySPRING improves dense reconstruction and prediction accuracy over PhysTwin, while reduced models retain stable physical and visual fidelity with up to a 2.30 times speed-up. We further demonstrate the effectiveness of PhySPRING in a Real2Sim robot policy-evaluation pipeline, where the reduced models are substituted zero-shot into ACT and $π_0$ evaluations, maintaining comparable manipulation success rates across downsampling levels while improving action-sampling effectiveness. Together, PhySPRING enables efficient and structure-preserving spring--mass reduction without sacrificing fidelity or robotic utility.

Elastic ribbons, slender structures whose length ($L$), width ($W$), and thickness ($b$) satisfy $L \gg W \gg b$, exhibit mechanical behaviors intermediate between one-dimensional rods ($L \gg W, b$) and two-dimensional plates ($L, W \gg b$). In quadratic Kirchhoff-type rod-based frameworks, such as Discrete Elastic Rods (DER), the governing equilibrium equations are independent of width, and therefore these models cannot capture width-dependent mechanical effects. Reduced centerline-based ribbon models attempt to capture width dependence via coupled bending-twisting energies. However, their relative accuracy remain unclear due to the absence of a unified simulation framework. In this work, we formulate a framework grounded in discrete differential geometry where the energy is expressed as functions of coupled bending-twisting strain measures along the centerline, rather than a linear sum of quadratic bending and twisting energies in DER. We derive analytical gradients and Hessians of the energy that enable implicit time integration. Within this unified setting, we compare five ribbon models: Kirchhoff, Sadowsky, Wunderlich, Sano, and Audoly. As a benchmark, a straight ribbon is longitudinally constrained into a pre-buckled arch and subjected to transverse displacement, inducing a supercritical pitchfork bifurcation. Predicted bifurcation thresholds are compared against shell-based finite element simulations, with the Sano model providing the closest agreement in capturing width-dependent shifts. Our high-performance JAX-based implementation achieves $\mathcal{O}(N)$ per-iteration cost and also confirms that Sano model introduces negligible per-iteration overhead relative to standard DER.

Tactile perception is key to dexterous manipulation, yet simulating high-resolution elastomer deformation remains computationally prohibitive. Finite element methods (FEM) deliver high fidelity but demand costly remeshing, while Material Point Methods (MPM) suffer from heavy particle-memory tradeoffs. We propose a {reduced-order neural simulation framework} that couples coarse-grained MPM dynamics with an implicit neural decoder to reconstruct sub-particle tactile details from compact latent states. The framework learns a continuous deformation manifold from paired high- and low-resolution simulations, enabling physically consistent, differentiable inference. Compared to the TacIPC, our method achieves over 65% faster simulation and {40% lower memory usage}, while maintaining better geometric fidelity. In tactile rendering and 3D surface reconstruction, our methods further improve accuracy by 25% and produce realistic depth images and surface mesh within a faster inference speed. These results demonstrate that the proposed reduced-order neural model enables high-detail, physically grounded tactile simulation with substantial efficiency gains for robotic interaction and optimization.

High-fidelity simulations such as computational fluid dynamics and finite element analysis are essential for modelling complex engineering systems but remain prohibitively expensive for tasks including parametric studies, optimisation and real-time control. Projection-based reduced-order models (ROMs) alleviate this cost by projecting full-order dynamics onto low-dimensional subspaces. On the other hand, projection-based ROMs can become less robust under parameter variation, and one approach is to adapt the projection basis (that is, the subspace) as the parameters change. We propose constrained extreme gradient boosting (cXGBoost) for predicting the proper orthogonal decomposition (POD) basis at given parameter settings. In particular, given the observed parameter-subspace pairs, we map the subspaces from the Grassmann manifold to a Euclidean space. Then, cXGBoost is trained in this Euclidean space with a constraint that the norm of the predicted vectors is bounded to ensure the injectivity of the mapping between the Euclidean space and the Grassmann manifold. On four parametric problems, a cylinder wake, Kolmogorov flow, the Kuramoto-Sivashinsky equation and a forced beam, cXGBoost has lower mean reconstruction errors than existing approaches, namely the projected Gaussian process and Grassmann interpolation.

Grid-based neural networks such as convolutional autoencoders are widely used in dimension reduction-based surrogate models for computational fluid dynamics. In recent years, the use of coordinate-based approaches like conditional neural fields has emerged. Their independence of the spatial discretization is a beneficial feature for various applications in computational fluid dynamics. This paper discusses the spatio-temporal prediction of aircraft ditching loads using a conditional neural field approach. The model is evaluated using two datasets for the dynamic loads of the fuselage of a DLR-D150 aircraft, one of which relates to a single fixed spatial discretization and the other that includes data from different discretizations. When paired with a long short-term memory (LSTM) network in the latent space, the neural field-based model achieves a spatio-temporal prediction accuracy for the first data set that is close to that of grid-dependent convolutional autoencoder-based models, and with significantly less parameters. Results for the second data set demonstrate the ability of the neural field-based approach to reconstruct ditching loads accurately for heterogeneous spatial discretizations. This allows for flexible use of training datasets generated for different geometries and/or discretizations, as well as the use of the surrogate model to predict loads for different configurations.

We introduce HAML (Hamiltonian Adaptation via Meta-Learning), a framework for fast online adaptation of effective Hamiltonian models of superconducting quantum processors. HAML proceeds in two phases. A supervised training phase uses an ensemble of simulated devices to learn an offline map from control inputs and device parameters to effective Hamiltonian coefficients. An online adaptation phase then uses a small number of hardware-accessible measurements to identify the unknown parameters of a new device. By training directly against effective two-qubit coefficients extracted from full multi-mode simulations, HAML implicitly learns the reduction from full multi-mode Hamiltonians to effective qubit descriptions without invoking perturbation theory. We further show that a variance-maximizing greedy selection of measurement configurations boosts online adaptation efficiency. We demonstrate HAML on a transmon-coupler-transmon system, recovering effective two-qubit coefficients across a wide range of operating regimes, including parameter regions where Schrieffer-Wolff perturbation theory (SWPT) breaks down. This establishes a scalable, sample-efficient approach to Hamiltonian reduction and characterization for near-term quantum processors, with direct implications for calibration, control, and error mitigation.

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on three systems of increasing complexity: an analytically tractable Cox--Ingersoll--Ross diffusion, a two-dimensional nonequilibrium diffusion with affine multiplicative noise, and a periodic soft-spin stochastic Landau--Lifshitz chain. Across these tests, the inferred models preserve the invariant statistics while reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.

This is our fourth work in the series on machine learning (ML) moment closure models for the radiative transfer equation (RTE). In the first three papers of this series, we considered the RTE in slab geometry in 1D1V (i.e. one dimension in physical space and one dimension in angular space), and introduced a gradient-based ML moment closure [1], then enforced the hyperbolicity through a symmetrizer [2], or together with physical characteristic speeds by learning the eigenvalues of the Jacobian matrix [3]. Here, we extend our framework to the RTE in 2D2V (i.e. two dimensions in physical space and two dimensions in angular space). The main idea is to preserve the leading part of the classical $P_N$ model and modify only the highest-order block row. By analyzing the structural properties of the $P_N$ model, we show that its coefficient matrices are symmetric and admit a block-tridiagonal structure. Then we use this property to introduce a block-diagonal symmetrizer for the ML moment model and derive explicit algebraic conditions on the closure blocks which guarantee the symmetrizable hyperbolicity of the resulting ML system. These conditions lead to a natural parametrization of the closure in terms of a symmetric positive definite matrix together with symmetric closure blocks, which can be learned from data while automatically enforcing symmetrizable hyperbolicity by construction. The numerical results show that the proposed framework improves upon the classical $P_N$ model while maintaining hyperbolicity.

Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic constraints exactly and reduces fast dynamics to their quasi-steady-state values in a single differentiable solve. Embedded in a physics-informed DeepONet, the layer recovers all fast and algebraic states exactly from slow-state predictions, removes the per-window stiffness-amplification pathway, and yields a stiffness-scaled Implicit Function Theorem gradient absent from penalty methods. Cascaded implicit layers extend this to multi-component systems with provable convergence. On a grid-forming inverter (stiffness ratio of about 4712), extended Newton attains 1.42% error versus 39.3% (penalty) and 57.0% (standard Newton); augmented Lagrangian and feedback linearization diverged. Two independently trained models compose without retraining (0.72% to 1.16% error, exact constraint satisfaction). Cross-domain validation on the Robertson stiff DAE (stiffness ratio up to $10^5$) confirms generalization. Conformal prediction provides 90% coverage with automatic out-of-distribution detection.

Reduced-order models are powerful for analyzing and controlling high-dimensional dynamical systems. Yet constructing these models for complex hybrid systems such as legged robots remains challenging. Classical approaches rely on hand-designed template models (e.g., LIP, SLIP), which, though insightful, only approximate the underlying dynamics. In contrast, data-driven methods can extract more accurate low-dimensional representations, but it remains unclear when stability and safety properties observed in the latent space meaningfully transfer back to the full-order system. To bridge this gap, we introduce HALO (Hybrid Auto-encoded Locomotion), a framework for learning latent reduced-order models of periodic hybrid dynamics directly from trajectory data. HALO employs an autoencoder to identify a low-dimensional latent state together with a learned latent Poincaré map that captures step-to-step locomotion dynamics. This enables Lyapunov analysis and the construction of an associated region of attraction in the latent space, both of which can be lifted back to the full-order state space through the decoder. Experiments on a simulated hopping robot and full-body humanoid locomotion demonstrate that HALO yields low-dimensional models that retain meaningful stability structure and predict full-order region-of-attraction boundaries.

We present parameter-interpolated dynamic mode decomposition (piDMD), a parametric reduced-order modeling framework that embeds known parameter-affine structure directly into the DMD regression step. Unlike existing parametric DMD methods which interpolate modes, eigenvalues, or reduced operators and can be fragile with sparse training data or multi-dimensional parameter spaces, piDMD learns a single parameter-affine Koopman surrogate reduced order model (ROM) across multiple training parameter samples and predicts at unseen parameter values without retraining. We validate piDMD on fluid flow past a cylinder, electron beam oscillations in transverse magnetic fields, and virtual cathode oscillations -- the latter two being simulated using an electromagnetic particle-in-cell (EMPIC) method. Across all benchmarks, piDMD achieves accurate long-horizon predictions and improved robustness over state-of-the-art interpolation-based parametric DMD baselines, with less training samples and with multi-dimensional parameter spaces.

Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear mappings, and utilize GPUs. Existing approaches typically learn evolution either on the full solution field, which requires capturing long-range spatial interactions at high computational cost, or on compressed latent representations obtained from autoencoders, which reduces the cost but often yields latent vectors that are difficult to evolve, since they primarily encode spatial information. Moreover, in parametric PDEs, the initial condition alone is not sufficient to determine the trajectory, and most current approaches are not evaluated on jointly predicting multiple solution components with differing magnitudes and parameter sensitivities. To address these challenges, we propose a joint model consisting of a convolutional encoder, a transformer operating on latent representations, and a decoder for reconstruction. The main novelties are joint training with multi-stage parameter injection and coordinate channel injection. Parameters are injected at multiple stages to improve conditioning. Physical coordinates are encoded to provide spatial information. This allows the model to dynamically adapt its computations to the specific PDE parameters governing each system, rather than learning a single fixed response. Experiments on the Advection-Diffusion-Reaction equation and Navier-Stokes flow around the cylinder wake demonstrate that our approach combines the efficiency of latent evolution with the fidelity of full-field models, outperforming DL-ROMs, latent transformers, and plain ViTs in multi-field prediction, reducing the relative rollout error by approximately $5$ times.

Model-free deep reinforcement learning (DRL) methods suffer from poor sample efficiency. To overcome this limitation, this work introduces an adaptive reduced-order-model (ROM)-based reinforcement learning framework for active flow control. In contrast to conventional actor--critic architectures, the proposed approach leverages a ROM to estimate the gradient information required for controller optimization. The design of the ROM structure incorporates physical insights. The ROM integrates a linear dynamical system and a neural ordinary differential equation (NODE) for estimating the nonlinearity in the flow. The parameters of the linear component are identified via operator inference, while the NODE is trained in a data-driven manner using gradient-based optimization. During controller--environment interactions, the ROM is continuously updated with newly collected data, enabling adaptive refinement of the model. The controller is then optimized through differentiable simulation of the ROM. The proposed ROM-based DRL framework is validated on two canonical flow control problems: Blasius boundary layer flow and flow past a square cylinder. For the Blasius boundary layer, the proposed method effectively reduces to a single-episode system identification and controller optimization process, yet it yields controllers that outperform traditional linear designs and achieve performance comparable to DRL approaches with minimal data. For the flow past a square cylinder, the proposed method achieves superior drag reduction with significantly fewer exploration data compared with DRL approaches. The work addresses a key component of model-free DRL control algorithms and lays the foundation for designing more sample-efficient DRL-based active flow controllers.

Precision reducers couple contact geometry, bearing support, structural deformation, and loading history. This paper presents a dynamic toolkit connecting distributed local contacts to the complete mechanical reaction path. Work-conjugate maps transfer displacement, reaction, and tangent contributions between contacts and rigid or reduced coordinates, with explicit allocation of contact and body elasticity. An implicit generalized-alpha solution distinguishes trial evaluations from accepted history. Contact records then support performance protocols and configured geometric or constitutive feedback. The numerical studies focus on mechanical coupling and pressure recovery. An idealized annular housing retains physical interfaces while its structural coordinates are reduced. Two reductions with similar static errors have cross-port response errors of 24.975 and 0.312 percent over the same frequency band relative to a common parent model. In a shared-pin example, a 20 micrometer radial displacement of one wheel changes the load on a second, fixed wheel by approximately 75 N. Removing cross-station compliance removes this incremental transfer on the tested sleeve-seating branch. A double-wheel cycloidal assembly relates torsional branch response to aggregate contact-load variation and normalized pressure fields. The pressure maxima remain sensitive to resolution despite small discrete force residuals. Further formulations specify how motion and contact records support precision, vibration, heat, wear, and durability models with their required inputs. The framework separates model representation and numerical resolution while retaining common definitions of motion, force, and observation.

Direct numerical simulations of turbulent reacting flows involving millions of grid points and detailed chemical mechanisms with hundreds of species and thousands of reactions are computationally prohibitive. To address this challenge, we present two data-driven chemical mechanism reduction formulations based on graph neural networks (GNNs) with message-passing transformer layers that learn nonlinear dependencies among species and reactions. The first formulation, GNN-SM, employs a pre-trained surrogate model to guide reduction across a broad range of reactor conditions. The second formulation, GNN-AE, uses an autoencoder formulation to obtain highly compact mechanisms that remain accurate within the thermochemical regimes used during training. The approaches are demonstrated on detailed mechanisms for methane (53 species, 325 reactions), ethylene (96 species, 1054 reactions), and iso-octane (1034 species, 8453 reactions). GNN-SM achieves reductions comparable to the established graph-based method DRGEP while maintaining accuracy across a wide range of thermochemical states. In contrast, GNN-AE achieves up to 95% reduction in species and reactions and outperforms DRGEP within its target conditions. Overall, the proposed framework provides an automated, machine-learning-based pathway for chemical mechanism reduction that can complement traditional expert-guided analytical approaches.

Task-driven design of soft robots requires models that are physically accurate and computationally efficient, while remaining transferable across actuator designs and task scenarios. However, existing modeling approaches typically face a fundamental trade-off between physical fidelity and computational efficiency, which limits model reuse across design and task variations and constrains scalable task-driven optimization. This paper presents a unified reduced-order finite element method (FEM)-based surrogate modeling pipeline for generalized task-driven soft robot design. High-fidelity FEM simulations characterize actuator behavior at the modular level, from which compact surrogate joint models are constructed for evaluation within a pseudo-rigid body model (PRBM). A meta-model maps actuator design parameters to surrogate representations, enabling rapid instantiation across a parameterized actuator family. The resulting models are embedded into a PRBM-based simulation environment, supporting task-level simulation and optimization under realistic physical constraints. The proposed pipeline is validated through sim-to-real transfer across multiple actuator types, including bellow-type pneumatic actuators and a tendon-driven soft finger, as well as two task-driven design studies: soft gripper co-design via Reinforcement Learning (RL) and 3D actuator shape matching via evolutionary optimization. The results demonstrate high accuracy, efficiency, and reliable reuse, providing a scalable foundation for autonomous task-driven soft robot design.

We investigate model order reduction (MOR) strategies for simulating unsteady hemodynamics within cerebrovascular systems, contrasting a physics-based intrusive approach with a data-driven non-intrusive framework. High-fidelity 3D Computational Fluid Dynamics (CFD) snapshots of an idealised basilar artery bifurcation are first compressed into a low-dimensional latent space using Proper Orthogonal Decomposition (POD). We evaluate the performance of a POD-Galerkin (POD-G) model, which projects the Navier-Stokes equations onto the reduced basis, against a POD-Reservoir Computing (POD-RC) model that learns the temporal evolution of coefficients through a recurrent architecture. A multi-harmonic and multi-amplitude training signal is introduced to improve training efficiency. Both methodologies achieve computational speed-ups on the order of 10^2 to 10^3 compared to full-order simulations, demonstrating their potential as efficient and accurate surrogates for predicting flow quantities such as wall shear stress.

Climate change and the rapid growth of urban populations are intensifying environmental stresses within cities, making the behavior of urban atmospheric flows a critical factor in public health, energy use, and overall livability. This study targets to develop fast and accurate models of urban pollutant dispersion to support decision-makers, enabling them to implement mitigation measures in a timely and cost-effective manner. To reach this goal, an end-to-end data-driven approach is proposed to model and predict the airflow and pollutant dispersion in a street canyon in skimming flow regime. A series of time-resolved snapshots obtained from large eddy simulation (LES) serves as the database. The proposed framework is based on four fundamental steps. Firstly, a reduced basis is obtained by spectral proper orthogonal decomposition (SPOD) of the database. The projection of the time series snapshot data onto the SPOD modes (time-domain approach) provides the temporal coefficients of the dynamics. Secondly, a nonlinear compression of the temporal coefficients is performed by autoencoder to reduce further the dimensionality of the problem. Thirdly, a reduced-order model (ROM) is learned in the latent space using Long Short-Term Memory (LSTM) netowrks. Finally, the pollutant dispersion is estimated from the predicted velocity field through convolutional neural network that maps both fields. The results demonstrate the efficacy of the model in predicting the instantaneous as well as statistically stationary fields over long time horizon.

Piecewise-linear nonlinear systems appear in many engineering disciplines. Prediction of the dynamic behavior of such systems is of great importance from practical and theoretical viewpoint. In this paper, a data-driven model order reduction method for piecewise-linear systems is proposed, which is based on dynamic mode decomposition (DMD). The overview of the concept of DMD is provided, and its application to model order reduction for nonlinear systems based on Galerkin projection is explained. The proposed approach uses impulse responses of the system to obtain snapshots of the state variables. The snapshots are then used to extract the dynamic modes that are used to form the projection basis vectors. The dynamics described by the equations of motion of the original full-order system are then projected onto the subspace spanned by the basis vectors. This produces a system with much smaller number of degrees of freedom (DOFs). The proposed method is applied to two representative examples of piecewise linear systems: a cantilevered beam subjected to an elastic stop at its end, and a bonded plates assembly with partial debonding. The reduced order models (ROMs) of these systems are constructed by using the Galerkin projection of the equation of motion with DMD modes alone, or DMD modes with a set of classical constraint modes to be able to handle the contact nonlinearity efficiently. The obtained ROMs are used for the nonlinear forced response analysis of the systems under harmonic loading. It is shown that the ROMs constructed by the proposed method produce accurate forced response results.

Stiff dynamical systems present a challenge for machine-learning reduced-order models (ML-ROMs), as explicit time integration becomes unstable in stiff regimes while implicit integration within learning loops is computationally expensive and often degrades training efficiency. Time reparameterization (TR) offers an alternative by transforming the independent variable so that rapid physical-time transients are spread over a stretched-time coordinate, enabling stable explicit integration on uniformly sampled grids. Although several TR strategies have been proposed, their effect on learnability in ML-ROMs remains incompletely understood. This work investigates time reparameterization as a stiffness-mitigation mechanism for neural ODE reduced-order modeling and introduces a trajectory-optimized TR (TOTR) formulation. The proposed approach casts time reparameterization as an optimization problem in arc-length coordinates, in which a traversal-speed profile is selected to penalize acceleration in stretched time. By targeting the smoothness of the training dynamics, this formulation produces reparameterized trajectories that are better conditioned and easier to learn than existing TR methods. TOTR is evaluated on three stiff problems: a parameterized stiff linear system, the van der Pol oscillator, and the HIRES chemical kinetics model. Across all cases, the proposed approach yields smoother reparameterizations and improved physical-time predictions under identical training regimens than other TR approaches. Quantitative results demonstrate loss reductions of one to two orders of magnitude compared to benchmark algorithms. These results highlight that effective stiffness mitigation in ML-ROMs depends critically on the regularity and learnability of the time map itself, and that optimization-based TR provides a robust framework for explicit reduced-order modeling of multiscale dynamical systems.

A novel convolutional autoencoder and neural ODE (CAE-NODE) framework is proposed for a reduced-order model (ROM) applied to transient 2D counterflow flames, as an extension of AE-NODE methods in homogeneous reactive systems to spatially resolved flows. The multidimensional thermochemical fields (256 x 256 grid, 21 variables) obtained from direct numerical simulations (DNS) are used in training the CAE, where convolutional layers learned the underlying spatial correlations, allowing the CAE to construct an unsupervised 3D latent manifold that is physically meaningful, smooth, and continuous in time. This results in a compression ratio of over 400,000 times. The NODE then subsequently learns the continuous-time dynamics on the latent manifold, enabling the prediction of the full temporal evolution of the flames by integrating forward in time from an initial condition. The results demonstrate that the CAE-NODE can accurately capture the entire transient process, including ignition, flame propagation, and the gradual transition to a non-premixed condition, with excellent agreement with the DNS, while adhering to conservation principles at virtually no computational cost compared to the reference DNS. Predictions remain accurate at strain rates outside the training range. Moreover, despite being unsupervised, the learned latent manifold is highly correlated with the flame-state descriptors such as the progress variable, mixture fraction, and the scalar dissipation rate. This study, for the first time, highlights the potential of CAE-NODE for surrogate modeling of unsteady dynamics of multi-dimensional reacting flows.

Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of parameter-dependent high-dimensional dynamical systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose an invertible AE architecture, named inv-AE, that computationally improves upon the stagnation of the reconstruction error typical of traditional AE architectures. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a 1-dimensional Burgers' equation, a 2-dimensional fluid flow around an obstacle with variable geometry, and a 3-dimensional Korteweg-de Vries, we show that (i) inv-AE mitigates the issue of the characteristic plateau of AEs and (ii) inv-AE can be combined with popular autoencoder-based ROM approaches, e.g., DL-ROM, to improve their accuracy.

Surrogate models provide compact relations between user-defined input parameters and output quantities of interest, enabling the efficient evaluation of complex parametric systems in many-query settings. Such capabilities are essential in a wide range of applications, including optimisation, control, data assimilation, uncertainty quantification, and emerging digital twin technologies in various fields such as manufacturing, personalised healthcare, smart cities, and sustainability. This article reviews established methodologies for constructing surrogate models exploiting either knowledge of the governing laws and the dynamical structure of the system (physics-based) or experimental observations (data-driven), as well as hybrid approaches combining these two paradigms. By revisiting the design of a surrogate model as a functional approximation problem, existing methodologies are reviewed in terms of the choice of (i) a reduced basis and (ii) a suitable approximation criterion. The paper reviews methodologies pertaining to the field of Scientific Machine Learning, and it aims at synthesising established knowledge, recent advances, and new perspectives on: dimensionality reduction, physics-based, and data-driven surrogate modelling based on proper orthogonal decomposition, proper generalised decomposition, and artificial neural networks; multi-fidelity methods to exploit information from sources with different fidelities; adaptive sampling, enrichment, and data augmentation techniques to enhance the quality of surrogate models.

Non-prehensile planar manipulation, including pushing and press-and-slide, is critical for diverse robotic tasks, but notoriously challenging due to hybrid contact mechanics, under-actuation, and asymmetric friction limits that traditionally necessitate computationally expensive iterative control. In this paper, we propose a mode-aware framework for planar manipulation with one or two robotic arms based on contact topology selection and reduced-order kinematic modeling. Our core insight is that complex wrench-twist limit surface mechanics can be abstracted into a discrete library of physically intuitive models. We systematically map various single-arm and bimanual contact topologies to simple non-holonomic formulations, e.g. unicycle for simplified press-and-slide motion. By anchoring trajectory generation to these reduced-order models, our framework computes the required object wrench and distributes feasible, friction-bounded contact forces via a direct algebraic allocator. We incorporate manipulator kinematics to ensure long-horizon feasibility and demonstrate our fast, optimization-free approach in simulation across diverse single-arm and bimanual manipulation tasks. Supplementary videos and additional information are available at: https://sites.google.com/view/pushpressslide

Magnetohydrodynamic (MHD) effects play a key role in the design and operation of nuclear fusion systems, where electrically conducting fluids (such as liquid metals or molten salts in reactor blankets) interact with magnetic fields of varying intensity and orientation, which affect the resulting flow. The numerical resolution of MHD models involves highly nonlinear multiphysics systems of equations and can become computationally expensive, particularly in multi-query, parametric, or real-time contexts. This work investigates a fully data-driven framework for MHD state reconstruction that combines dimensionality reduction via Singular Value Decomposition (SVD) with the SHallow REcurrent Decoder (SHRED), a neural network architecture designed to recover the full spatio-temporal state from sparse time-series measurements of a limited number of observables. The methodology is applied to a parametric MHD test case involving compressible lead-lithium flow in a stepped channel subjected to thermal gradients and magnetic fields spanning a broad range of intensities. To improve efficiency, the full-order dataset is first compressed using SVD, yielding a reduced representation used as reference truth for training. Only temperature measurements from three sensors are provided as input, while the network reconstructs the full fields of velocity, pressure, and temperature. To assess robustness with respect to sensor placement, thirty randomly generated sensor configurations are tested in ensemble mode. Results show that SHRED accurately reconstructs the full MHD state even for magnetic field intensities not included in the training set. These findings demonstrate the potential of SHRED as a computationally efficient surrogate modeling strategy for fusion-relevant multiphysics problems, enabling low-cost state estimation with possible applications in real-time monitoring and control.

We propose neural network operator inference (NN-OpInf): a structure-preserving, composable, and minimally restrictive operator inference framework for the non-intrusive reduced-order modeling of dynamical systems. The approach learns latent dynamics from snapshot data, enforcing local operator structure such as skew-symmetry, (semi-)positive definiteness, and gradient preservation, while also reflecting complex dynamics by supporting additive compositions of heterogeneous operators. We present practical training strategies and analyze computational costs relative to linear and quadratic polynomial OpInf (P-OpInf). Numerical experiments across several nonlinear and parametric problems demonstrate improved accuracy, stability, and robustness over P-OpInf and prior NN-ROM formulations, particularly when the dynamics are not well represented by polynomial models. These results suggest that NN-OpInf can serve as an effective drop-in replacement for P-OpInf when the dynamics to be modeled contain non-polynomial nonlinearities, offering potential gains in accuracy and out-of-distribution performance at the expense of higher training computational costs and a more difficult, non-convex learning problem.

We investigate geometric regularization strategies for learned latent representations in encoder--decoder reduced-order models. In a fixed experimental setting for the advection--diffusion--reaction (ADR) equation, we model latent dynamics using a neural ODE and evaluate four regularization approaches applied during autoencoder pre-training: (a) near-isometry regularization of the decoder Jacobian, (b) a stochastic decoder gain penalty based on random directional gains, (c) a second-order directional curvature penalty, and (d) Stiefel projection of the first decoder layer. Across multiple seeds, we find that (a)--(c) often produce latent representations that make subsequent latent-dynamics training with a frozen autoencoder more difficult, especially for long-horizon rollouts, even when they improve local decoder smoothness or related sensitivity proxies. In contrast, (d) consistently improves conditioning-related diagnostics of the learned latent dynamics and tends to yield better rollout performance. We discuss the hypothesis that, in this setting, the downstream impact of latent-geometry mismatch outweighs the benefits of improved decoder smoothness.