Reduced order modeling with shallow recurrent decoder networks

Cost analysis of debugging systems : B. P. Lester. Mass. Inst. of Tech. Electrical Engng. Dept. U.S.A. (1101–7203) T72-00944 MAC-TR-90, N-0014-70-A-0362-0001, September (1971), pp. 115

Fast yet accurate prediction of the resin flow behavior within the designed fiber layup is crucial for planning the manufacturing of composites using resin transfer moulding (RTM). A key challenge in RTM is ensuring complete saturation of the layup during infusion, which could be hindered by the local variation of permeability - thus creating complex flow patterns and defects such as dry spots and/or voids in the infused component. Current high-fidelity numerical modelling workflows for predicting permeability suffer from large computation times and only allow simulations investigating a few fiber layers within the full layup. The current paper aims to showcase a reduced-order model (ROM) based workflow to address these two limitations. The proposed workflow establishes a ROM by combining a design-of-experiments methodology with high-fidelity CFD simulations of micro-meso scale resin infusion. The validated high-fidelity simulations capture the dependence of the principal permeability components on both the intra and inter yarn geometric packing, while the ROM can predict the permeability based on established response functions relating the geometry to the final permeability. Both the high-fidelity micro-meso CFD model as well as the reduced order meso-scale permeability prediction model are subsequently coupled to a macro-scale resin infusion simulation that emulates a flat rectilinear flow within a mould with different local permeability values based on the local fiber layup. The implication of the predicted meso-scale permeability on the macro-scale infusion is studied, providing a comparative evaluation of the high-fidelity simulation and the ROM.

Design of complex cyber-physical products and systems such as aircrafts, cars, underwater vehicles, robotic arms, etc., is a lengthy process in which designers across multiple teams rely on well established engineering workflows executing simulations of various fidelities. These workflows usually implement, at some point in the design process, high-fidelity physics-based simulations that involve solving ordinary and/or partial differential equations (PDEs) in the different disciplines considered (e.g., fluid dynamics, structures, heat transfer, dynamics and controls etc.). While significant research is being conducted in developing algorithms to solve these PDEs faster, there is currently a reliance on data-driven reduced order modeling (ROM) to augment or replace the traditional simulations; however, these ROMs are mainly employed during the asset operation phase rather than their design. Additionally, the need for these simulation acceleration techniques is enforced by the need for mission optimized designs, for example in the case of unmanned aerial vehicles and unmanned underwater vehicles. With optimization becoming more prevalent in engineering, geometric changes are often enforced in the design exploration processes. Thus, it is essential for the ROMs to handle geometric changes. When considering ROMs, projection-based techniques rely on having a grid of points (i.e., mesh) which are projected into a common lower-dimensional manifold. In the presence of geometric changes, mesh morphing techniques have recently solved this issue provided modification of the geometry and mesh generation steps is possible; however, in many practical situations designers only have access to a black-box workflow whose steps cannot be edited or the mesh morphing skills needed are not immediately available or are simply prohibitive. Thus, we present a design-focused ROM which can be coupled with existing black-box computer-aided engineering workflows. We demonstrate the ability to develop ROMs in practical situations where designers have only access to a black-box workflow. Our proposed method converts varying geometry results from black-box workflows into a common dimension mesh which enables known ROM techniques to be applied during the product design phase. We apply these methods to accelerate fluid mechanics simulations and strength analysis in the design of unmanned underwater vehicles. For these design cases, the proposed method reconstructs the relevant fields with more than two order of magnitude speed-ups with minimal loss in accuracy.

Kramers' 1940 paper and its successive elaborations have extensively explored the transition rate between two stable situations, that is, in the language of system dynamics, the transition between the basins of attraction of two stable fixed point attractors. In a nonequilibrium system some of the above conditions may be violated, either because one of the two fixed points is unstable, as in the case of transient phenomena, or because both fixed points are unstable, as in the case of heteroclinic chaos, or because the attractors are more complex than fixed points, as in a chaotic dynamics where two or more strange attractors coexist. Furthermore, there is recent experimental evidence of space-time complexity consisting in the alternate or simultaneous oscillation of many modes, each one with its own (possibly chaotic) dynamics. In all the above cases, coexistence of many alternative paths implies a choice, either due to noise or self-triggered by the same interacting degrees of freedom. A review of the above phenomena in the case of nonequilibrium optical systems is here presented, with the aim of stimulating theoretical investigation on these novel rate processes.

The new approach for combinatorial optimization problems using chaotic dynamics is discussed. We show effectiveness of chaotic neuro dynamics for solving combinatorial optimization problems by applying the chaotic neural network to traveling salesman problems. In this paper, we adopt the chaotic neural network model with two internal states, corresponding to mutual interactions which minimize an energy function and refractoriness which induce chaotic dynamics. We investigate relationships between solving abilities and different model parameters such as decay parameters of two internal states, Lyapunov exponents and first order statistics of firing patterns.

This paper presents a physics-based data-driven method to learn predictive reduced-order models (ROMs) from high-fidelity simulations and illustrates it in the challenging context of a single-injector combustion process. The method combines the perspectives of model reduction and machine learning. Model reduction brings in the physics of the problem, constraining the ROM predictions to lie on a subspace defined by the governing equations. This is achieved by defining the ROM in proper orthogonal decomposition (POD) coordinates, which embed the rich physics information contained in solution snapshots of a high-fidelity computational fluid dynamics model. The machine learning perspective brings the flexibility to use transformed physical variables to define the POD basis. This is in contrast to traditional model reduction approaches that are constrained to use the physical variables of the high-fidelity code. Combining the two perspectives, the approach identifies a set of transformed physical variables that expose quadratic structure in the combustion governing equations and learns a quadratic ROM from transformed snapshot data. This learning does not require access to the high-fidelity model implementation. Numerical experiments show that the ROM accurately predicts temperature, pressure, velocity, species concentrations, and the limit-cycle amplitude, with speedups of more than five orders of magnitude over high-fidelity models. Our ROM simulation is shown to be predictive 200% past the training interval. ROM-predicted pressure traces accurately match the phase of the pressure signal and yield good approximations of the limit-cycle amplitude.

Addressing climate change through carbon capture and storage (CCS) technologies necessitates advanced computational methodologies for subsurface CO2 storage monitoring. This study focuses on the Illinois Basin Decatur Project (IBDP), a CCS demonstration pilot aimed at CO2 injection into a deep saline reservoir. We introduce a novel framework combining Dynamic Mode Decomposition (DMD), a data-driven model reduction technique, with direct data assimilation to streamline the calibration of CO2 plume evolution models. This approach enhances rapid tracking and overcomes the computational challenges of traditional high-fidelity numerical reservoir simulations known as the full-order model (FOM). Using DMD, we analyzed five distinct FOM simulation cases of the IBDP with varied permeability in the Mt. Simon section to develop reduced-order models (ROMs). These ROMs utilize three state variables: reservoir pressure, CO2 plume saturation, and bottom-hole pressure (BHP) from a CO2 injection well alongside multi-level pressures from a monitoring well derived from the FOM and the actual field data respectively. Initial FOM simulation cases assessed the impact of permeability multipliers on pressure responses. We then transformed these into ROMs using DMD, preserving essential dynamics. Linear interpolation between permeabilities and DMD outputs—modes and eigenvalues—established relationships for rapid BHP prediction under different scenarios. Employing a Kalman filter, we optimized a global permeability multiplier, using the ROMs, to align measured and simulated BHP values, ensuring model calibration. The final calibrated FOM was further decomposed to a DMD-based ROM, enabling quick, accurate predictions, significantly reducing computational time from hours to minutes. Utilizing an ROM derived through DMD, we achieved an order of 160 reduction in computational time (from 8 hours to just 3 minutes) for a 3-year historical CO2 injection period modeled with 547,000 cells of the FOM. The ROM demonstrated remarkable fidelity, with a mean absolute error of 1.46 psi for pressure and 3.7e-05 for CO2 plume saturation, effectively capturing the dynamics of the full-order model. This substantial decrease in computational time illustrates an advantageous trade-off between speed and accuracy, optimizing the potential for long-term forecasting and monitoring of CO2 sequestration. Incorporating the IBDP as a case study, this research contributes a significant advancement to reservoir simulation practices, offering a potent, efficient tool for CCS monitoring. By integrating DMD for ROM construction with precise data assimilation-based calibration, the study provides a comprehensive solution for swift and accurate subsurface CO2 plume tracking, essential for the successful implementation of CCS projects and the broader effort to mitigate climate change impacts.

Reduced Order Models (ROMs) are of considerable importance in many areas of engineering in which computational time presents difficulties. Established approaches employ projection-based reduction, such as Proper Orthogonal Decomposition. The limitation of the linear nature of such operators is typically tackled via a library of local reduction subspaces, which requires the assembly of numerous local ROMs to address parametric dependencies. Our work attempts to define a more generalisable mapping between parametric inputs and reduced bases for the purpose of generative modeling. We propose the use of Variational Autoencoders (VAEs) in place of the typically utilised clustering or interpolation operations, for inferring the fundamental vectors, termed as modes, which approximate the manifold of the model response for any and each parametric input state. The derived ROM still relies on projection bases, built on the basis of full-order model simulations, thus retaining the imprinted physical connotation. However, it additionally exploits a matrix of coefficients that relates each local sample response and dynamics to the global phenomena across the parametric input domain. The VAE scheme is utilised for approximating these coefficients for any input state. This coupling leads to a high-precision low-order representation, which is particularly suited for problems where model dependencies or excitation traits cause the dynamic behavior to span multiple response regimes. Moreover, the probabilistic treatment of the VAE representation allows for uncertainty quantification on the reduction bases, which may then be propagated to the ROM response. The performance of the proposed approach is validated on an open-source simulation benchmark featuring hysteresis and multi-parametric dependencies, and on a large-scale wind turbine tower characterised by nonlinear material behavior and model uncertainty.

This paper presents a physics-based data-driven method to learn predictive reduced-order models (ROMs) from high-fidelity simulations, and illustrates it in the challenging context of a single-injector combustion process. The method combines the perspectives of model reduction and machine learning. Model reduction brings in the physics of the problem, constraining the ROM predictions to lie on a subspace defined by the governing equations. This is achieved by defining the ROM in proper orthogonal decomposition (POD) coordinates, which embed the rich physics information contained in solution snapshots of a high-fidelity computational fluid dynamics (CFD) model. The machine learning perspective brings the flexibility to use transformed physical variables to define the POD basis. This is in contrast to traditional model reduction approaches that are constrained to use the physical variables of the high-fidelity code. Combining the two perspectives, the approach identifies a set of transformed physical variables that expose quadratic structure in the combustion governing equations and learns a quadratic ROM from transformed snapshot data. This learning does not require access to the high-fidelity model implementation. Numerical experiments show that the ROM accurately predicts temperature, pressure, velocity, species concentrations, and the limit-cycle amplitude, with speedups of more than five orders of magnitude over high-fidelity models. Moreover, ROM-predicted pressure traces accurately match the phase of the pressure signal and yield good approximations of the limit-cycle amplitude.

A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations

In this paper, a non-intrusive reduced-order model (ROM) for parametric reactor kinetics simulations is presented. Time-dependent ROMs are notoriously data intensive and difficult to implement when nonlinear multiphysics phenomena are considered. These challenges are exacerbated when parametric dependencies are included. The proper orthogonal decomposition mode coefficient interpolation (POD-MCI) ROM presented in this work can be constructed directly from lower-dimensional full-order model (FOM) outputs and is independent of the underlying model. This greatly alleviates the data requirement of many existing ROMs and can be used without modification on arbitrarily complex models or experimental data. The POD-MCI ROM is demonstrated on a number of examples and yields accurate characterizations of the parametric behaviors of both FOM outputs and derived quantities of interest within the selected parameter spaces, at extremely attractive computational speedup factors relative to FOMs.

Non-linear analysis is of increasing importance in wind energy engineering as a result of their exposure in extreme conditions and the ever-increasing size and slenderness of wind turbines. Whilst modern computing capabilities facilitate execution of complex analyses, certain applications which require multiple or real-time analyses remain a challenge, motivating adoption of accelerated computing schemes, such as reduced order modelling (ROM) methods. Soil structure interaction (SSI) simulations fall in this class of problems, with the non-linear restoring force significantly affecting the dynamic behaviour of the turbine. In this work, we propose a ROM approach to the SSI problem using a recently developed ROM methodology. We exploit a data-driven non-linear ROM methodology coupling an autoencoder with long short-term memory (LSTM) neural networks. The ROM is trained to emulate a steel monopile foundation constrained by non-linear soil and subject to forces and moments at the top of the foundation, which represent the equivalent loading of an operating turbine under wind and wave forcing. The ROM well approximates the time domain and frequency domain response of the Full Order Model (FOM) over a range of different wind and wave loading regimes, whilst reducing the computational toll by a factor of 300. We further propose an error metric for capturing isolated failure instances of the ROM.

Reduced-order flow modeling and geological parameterization for ensemble-based data assimilation

Fluid---structure interaction (FSI) phenomena are of interest in several engineering fields. It is highly desirable to develop computationally efficient models to predict the dynamics of FSI. The complexity of modeling lies in the highly non-linear response of both the fluid and structure. The current study proposes an overall model containing two blocks corresponding to a force model and a structural model. The force model consists of two submodels: one for the amplitude and one for the frequency, where the latter is composed of an input/output linear model and a non-linear corrector. The amplitude submodel and the non-linear corrector term in the frequency submodel are modeled using an Hammerstein---Wiener modeling technique in which the non-linear input and output functions are determined by training neural networks using a training dataset. The current model is tested on a well-known fluid---structure interaction problem: a suspended rigid cylinder immersed in a flow at a low Reynolds number regime that exhibits a non-linear behavior. First, a training dataset is generated for a given input profile using a high-fidelity numerical simulation and it is used to train the reduced-order model. Subsequently, the trained model is given a different input profile (i.e., a validation profile) to compare its predictive capability against the high-fidelity numerical simulation. The validation profile is significantly different from the one used for training. The predictive performance of the current reduced-order model is further compared with the results obtained from a reduced-order model that uses polynomial fitting. We demonstrate that the current model provides a superior performance for the validation profile, i.e., it results in a better prediction.

Chaotic solutions in dynamic linear programming