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1. A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients

   https://doi.org/10.1016/j.cma.2023.116684  

   Chen, Peiyi; Hu, Tianchen; Guilleminot, Johann (February 2024, Computer Methods in Applied Mechanics and Engineering)

   Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov 𝑛-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN- based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy. 

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2. Concurrent multiscale simulations of nonlinear random materials using probabilistic learning

   https://doi.org/10.1016/j.cma.2024.116837  

   Chen, Peiyi; Guilleminot, Johann; Soize, Christian (March 2024, Computer Methods in Applied Mechanics and Engineering)

   Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov n-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN- based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy. 

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3. Waveform inversion via reduced order modeling

   https://doi.org/10.1190/GEO2022-0070.1  

   Borcea, Liliana; Garnier, Josselin; Mamonov, Alexander V.; Zimmerling, Jörn (March 2023, GEOPHYSICS)

   We introduce a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, we find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minima. 

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4. A Non-Convex Approach To Joint Sensor Calibration And Spectrum Estimation

   https://doi.org/10.1109/SSP.2018.8450691  

   Cho, Myung; Liao, Wenjing; Chi, Yuejie (June 2018, 2018 IEEE Statistical Signal Processing Workshop (SSP))

   Blind sensor calibration for spectrum estimation is the problem of estimating the unknown sensor calibration parameters as well as the parameters-of-interest of the impinging signals simultaneously from snapshots of measurements obtained from an array of sensors. In this paper, we consider blind phase and gain calibration (BPGC) problem for direction-of-arrival estimation with multiple snapshots of measurements obtained from an uniform array of sensors, where each sensor is perturbed by an unknown gain and phase parameter. Due to the unknown sensor and signal parameters, BPGC problem is a highly nonlinear problem. Assuming that the sources are uncorrelated, the covariance matrix of the measurements in a perfectly calibrated array is a Toeplitz matrix. Leveraging this fact, we first change the nonlinear problem to a linear problem considering certain rank-one positive semidefinite matrix, and then suggest a non-convex optimization approach to find the factor of the rank-one matrix under a unit norm constraint to avoid trivial solutions. Numerical experiments demonstrate that our proposed non-convex optimization approach provides better or competitive recovery performance than existing methods in the literature, without requiring any tuning parameters. 

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5. Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

   https://doi.org/10.3390/fluids5010039  

   Xie, Xuping; Webster, Clayton; Iliescu, Traian (March 2020, Fluids)

   null (Ed.)

   Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models. 

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