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We present a novel probabilistic approach for generating multi-fidelity data while accounting for errors inherent in both low- and high-fidelity data. In this approach a graph Laplacian constructed from the low-fidelity data is used to define a multivariate Gaussian prior density for the coordinates of the true data points. In addition, few high-fidelity data points are used to construct a conjugate likelihood term. Thereafter, Bayes rule is applied to derive an explicit expression for the posterior density which is also multivariate Gaussian. The maximum \textit{a posteriori} (MAP) estimate of this density is selected to be the optimal multi-fidelity estimate. It is shown that the MAP estimate and the covariance of the posterior density can be determined through the solution of linear systems of equations. Thereafter, two methods, one based on spectral truncation and another based on a low-rank approximation, are developed to solve these equations efficiently. The multi-fidelity approach is tested on a variety of problems in solid and fluid mechanics with data that represents vectors of quantities of interest and discretized spatial fields in one and two dimensions. The results demonstrate that by utilizing a small fraction of high-fidelity data, the multi-fidelity approach can significantly improve the accuracy of a large collection of low-fidelity data points.

We propose a framework to perform Bayesian inference using conditional score-based diffusion models to solve a class of inverse problems in mechanics involving the inference of a specimen's spatially varying material properties from noisy measurements of its mechanical response to loading. Conditional score-based diffusion models are generative models that learn to approximate the score function of a conditional distribution using samples from the joint distribution. More specifically, the score functions corresponding to multiple realizations of the measurement are approximated using a single neural network, the so-called score network, which is subsequently used to sample the posterior distribution using an appropriate Markov chain Monte Carlo scheme based on Langevin dynamics. Training the score network only requires simulating the forward model. Hence, the proposed approach can accommodate black-box forward models and complex measurement noise. Moreover, once the score network has been trained, it can be re-used to solve the inverse problem for different realizations of the measurements. We demonstrate the efficacy of the proposed approach on a suite of high-dimensional inverse problems in mechanics that involve inferring heterogeneous material properties from noisy measurements. Some examples we consider involve synthetic data, while others include data collected from actual elastography experiments. Further, our applications demonstrate that the proposed approach can handle different measurement modalities, complex patterns in the inferred quantities, non-Gaussian and non-additive noise models, and nonlinear black-box forward models. The results show that the proposed framework can solve large-scale physics-based inverse problems efficiently.

Generative modeling has drawn much attention in creative and scientific data generation tasks. Score-based Diffusion Models, a type of generative model that iteratively learns to denoise data, have shown state-of-the-art results on tasks such as image generation, multivariate time series forecasting, and robotic trajectory planning. Using score-based diffusion models, this work implements a novel generative framework to generate ballistic transfers from Earth to Mars. We further analyze the model's ability to learn the characteristics of the original dataset and its ability to produce transfers that follow the underlying dynamics. Ablation studies were conducted to determine how model performance varies with model size and trajectory temporal resolution. In addition, a performance benchmark is designed to assess the generative model's usefulness for trajectory design, conduct model performance comparisons, and lay the groundwork for evaluating different generative models for trajectory design beyond diffusion. The results of this analysis showcase several useful properties of diffusion models that, when taken together, can enable a future system for generative trajectory design powered by diffusion models., Comment: To be presented at the 2024 Astrodynamics Specialist Conference

We propose a novel modular inference approach combining two different generative models -- generative adversarial networks (GAN) and normalizing flows -- to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows.

Increases in wildfire activity and the resulting impacts have prompted the development of high-resolution wildfire behavior models for forecasting fire spread. Recent progress in using satellites to detect fire locations further provides the opportunity to use measurements to improve fire spread forecasts from numerical models through data assimilation. This work develops a method for inferring the history of a wildfire from satellite measurements, providing the necessary information to initialize coupled atmosphere-wildfire models from a measured wildfire state in a physics-informed approach. The fire arrival time, which is the time the fire reaches a given spatial location, acts as a succinct representation of the history of a wildfire. In this work, a conditional Wasserstein Generative Adversarial Network (cWGAN), trained with WRF-SFIRE simulations, is used to infer the fire arrival time from satellite active fire data. The cWGAN is used to produce samples of likely fire arrival times from the conditional distribution of arrival times given satellite active fire detections. Samples produced by the cWGAN are further used to assess the uncertainty of predictions. The cWGAN is tested on four California wildfires occurring between 2020 and 2022, and predictions for fire extent are compared against high resolution airborne infrared measurements. Further, the predicted ignition times are compared with reported ignition times. An average Sorensen's coefficient of 0.81 for the fire perimeters and an average ignition time error of 32 minutes suggest that the method is highly accurate.

The solution of probabilistic inverse problems for which the corresponding forward problem is constrained by physical principles is challenging. This is especially true if the dimension of the inferred vector is large and the prior information about it is in the form of a collection of samples. In this work, a novel deep learning based approach is developed and applied to solving these types of problems. The approach utilizes samples of the inferred vector drawn from the prior distribution and a physics-based forward model to generate training data for a conditional Wasserstein generative adversarial network (cWGAN). The cWGAN learns the probability distribution for the inferred vector conditioned on the measurement and produces samples from this distribution. The cWGAN developed in this work differs from earlier versions in that its critic is required to be 1-Lipschitz with respect to both the inferred and the measurement vectors and not just the former. This leads to a loss term with the full (and not partial) gradient penalty. It is shown that this rather simple change leads to a stronger notion of convergence for the conditional density learned by the cWGAN and a more robust and accurate sampling strategy. Through numerical examples it is shown that this change also translates to better accuracy when solving inverse problems. The numerical examples considered include illustrative problems where the true distribution and/or statistics are known, and a more complex inverse problem motivated by applications in biomechanics., Comment: 34 pages, 9 figures, 3 tables, 1 appendix

Low-fidelity data is typically inexpensive to generate but inaccurate. On the other hand, high-fidelity data is accurate but expensive to obtain. Multi-fidelity methods use a small set of high-fidelity data to enhance the accuracy of a large set of low-fidelity data. In the approach described in this paper, this is accomplished by constructing a graph Laplacian using the low-fidelity data and computing its low-lying spectrum. This spectrum is then used to cluster the data and identify points that are closest to the centroids of the clusters. High-fidelity data is then acquired for these key points. Thereafter, a transformation that maps every low-fidelity data point to its bi-fidelity counterpart is determined by minimizing the discrepancy between the bi- and high-fidelity data at the key points, and to preserve the underlying structure of the low-fidelity data distribution. The latter objective is achieved by relying, once again, on the spectral properties of the graph Laplacian. This method is applied to a problem in solid mechanics and another in aerodynamics. In both cases, this methods uses a small fraction of high-fidelity data to significantly improve the accuracy of a large set of low-fidelity data.

These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics. The main objective of these notes is to introduce a student who is familiar with concepts in linear algebra and partial differential equations to select topics in deep learning. These lecture notes exploit the strong connections between deep learning algorithms and the more conventional techniques of computational physics to achieve two goals. First, they use concepts from computational physics to develop an understanding of deep learning algorithms. Not surprisingly, many concepts in deep learning can be connected to similar concepts in computational physics, and one can utilize this connection to better understand these algorithms. Second, several novel deep learning algorithms can be used to solve challenging problems in computational physics. Thus, they offer someone who is interested in modeling a physical phenomena with a complementary set of tools., Comment: 7 chapters

In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. This work describes a new architecture for operator networks that mimics the form of the numerical solution obtained from an approximate variational or weak formulation of the problem. The application of these ideas to a generic elliptic PDE leads to a variationally mimetic operator network (VarMiON). Like the conventional Deep Operator Network (DeepONet) the VarMiON is also composed of a sub-network that constructs the basis functions for the output and another that constructs the coefficients for these basis functions. However, in contrast to the DeepONet, the architecture of these sub-networks in the VarMiON is precisely determined. An analysis of the error in the VarMiON solution reveals that it contains contributions from the error in the training data, the training error, the quadrature error in sampling input and output functions, and a "covering error" that measures the distance between the test input functions and the nearest functions in the training dataset. It also depends on the stability constants for the exact solution operator and its VarMiON approximation. The application of the VarMiON to a canonical elliptic PDE and a nonlinear PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet and a recently proposed multiple-input operator network (MIONet). Further, its performance is more robust to variations in input functions, the techniques used to sample the input and output functions, the techniques used to construct the basis functions, and the number of input functions., Comment: 49 pages, 18 figures, 1 Appendix

In this work, we train conditional Wasserstein generative adversarial networks to effectively sample from the posterior of physics-based Bayesian inference problems. The generator is constructed using a U-Net architecture, with the latent information injected using conditional instance normalization. The former facilitates a multiscale inverse map, while the latter enables the decoupling of the latent space dimension from the dimension of the measurement, and introduces stochasticity at all scales of the U-Net. We solve PDE-based inverse problems to demonstrate the performance of our approach in quantifying the uncertainty in the inferred field. Further, we show the generator can learn inverse maps which are local in nature, which in turn promotes generalizability when testing with out-of-distribution samples.

Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is tackling their ill-posed nature. Bayesian inference provides a principled approach for overcoming this by formulating the inverse problem into a statistical framework. However, it is challenging to apply when inferring fields that have discrete representations of large dimensions (the so-called "curse of dimensionality") and/or when prior information is available only in the form of previously acquired solutions. In this work, we present a novel method for efficient and accurate Bayesian inversion using deep generative models. Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems. Our statistical framework preserves the underlying physics and is demonstrated to yield accurate results with reliable uncertainty estimates, even in the absence of information about underlying noise model, which is a significant challenge with many existing methods. We demonstrate the effectiveness of proposed method on a variety of inverse problems which include both synthetic as well as experimentally observed data., Comment: Paper: 38 pages, 12 figures, 3 Tables

The spread of infectious epidemics is often accelerated by super-spreader events. Understanding their effect is important, particularly in the context of standard epidemiological models, which require estimates for parameters such as $R_0$. In this letter, we show that the effective value of $R_0$ in super-spreader situations is significantly large, of the order of hundreds, suggesting a delta-function-like behavior during the event. Use of a well-mixed room model supports these findings. They elucidate infection kinetic modeling in enclosed environments, which differ from the standard SIR model, and provide expressions for $R_0$ in terms of physical and operational parameters. The overall impact of super-spreader events can be significant, depending on the state of the epidemic and how the infections generated by the event subsequently spread in the community.

Motivated by analogies between the spreading of human-to-human infections and of chemical processes, we develop a comprehensive model that accounts both for infection and for transport. In this analogy, the three different populations of infection models correspond to three chemical species. Areal densities emerge as the key variables, thus capturing the effect of spatial density. We derive expressions for the kinetics of the infection rates and for the important parameter R0, that include areal density and its spatial distribution. Coupled with mobility the model allows the study of various effects. We first present results for a batch reactor, the chemical process equivalent of the SIR model. Because density makes R0 a decreasing function of the process extent, the infection curves are different and smaller than for the standard SIR model. We show that the effect of the initial conditions is limited to the onset of the epidemic. We derive effective infection curves for a number of cases, including a back-and-forth commute between regions of low and high R0 environments. We then consider spatially distributed systems. We show that diffusion leads to traveling waves, which in 1-D geometries propagate at a constant speed and with a constant shape, both of which are sole functions of R0. The infection curves are slightly different than for the batch problem, as diffusion mitigates the infection intensity, thus leading to an effective lower R0. The dimensional wave speed is found to be proportional to the product of the square root of the diffusivity and of an increasing function of R0, confirming the importance of restricting mobility in arresting the propagation of infection. We examine the interaction of infection waves under various conditions and scenarios, and extend the wave propagation analysis to 2-D heterogeneous systems.

There is no consensus regarding the radiomic feature terminology, the underlying mathematics, or their implementation. This creates a scenario where features extracted using different toolboxes could not be used to build or validate the same model leading to a non-generalization of radiomic results. In this study, the image biomarker standardization initiative (IBSI) established phantom and benchmark values were used to compare the variation of the radiomic features while using 6 publicly available software programs and 1 in-house radiomics pipeline. All IBSI-standardized features (11 classes, 173 in total) were extracted. The relative differences between the extracted feature values from the different software and the IBSI benchmark values were calculated to measure the inter-software agreement. To better understand the variations, features are further grouped into 3 categories according to their properties: 1) morphology, 2) statistic/histogram and 3)texture features. While a good agreement was observed for a majority of radiomics features across the various programs, relatively poor agreement was observed for morphology features. Significant differences were also found in programs that use different gray level discretization approaches. Since these programs do not include all IBSI features, the level of quantitative assessment for each category was analyzed using Venn and the UpSet diagrams and also quantified using two ad hoc metrics. Morphology features earns lowest scores for both metrics, indicating that morphological features are not consistently evaluated among software programs. We conclude that radiomic features calculated using different software programs may not be identical and reliable. Further studies are needed to standardize the workflow of radiomic feature extraction., Comment: 21 pages, 8 figures

Bayesian inference is used extensively to quantify the uncertainty in an inferred field given the measurement of a related field when the two are linked by a mathematical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and/or have prior distributions that are difficult to characterize mathematically. In this work we demonstrate how the approximate distribution learned by a deep generative adversarial network (GAN) may be used as a prior in a Bayesian update to address both these challenges. We demonstrate the efficacy of this approach on two distinct, and remarkably broad, classes of problems. The first class leads to supervised learning algorithms for image classification with superior out of distribution detection and accuracy, and for image inpainting with built-in variance estimation. The second class leads to unsupervised learning algorithms for image denoising and for solving physics-driven inverse problems.

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore, sheds light on learning algorithms. This paper is devoted to the study of a parameterized family of divergence form elliptic operators that arise as the large data limit of graph Laplacians. The link between a three-parameter family of graph Laplacians and a three-parameter family of differential operators is explained. The spectral properties of these differential operators are analyzed in the situation where the data comprises two nearly separated clusters, in a sense which is made precise. In particular, we investigate how the spectral gap depends on the three parameters entering the graph Laplacian, and on a parameter measuring the size of the perturbation from the perfectly clustered case. Numerical results are presented which exemplify and extend the analysis: the computations study situations in which there are two nearly separated clusters, but which violate the assumptions used in our theory; situations in which more than two clusters are present, also going beyond our theory; and situations which demonstrate the relevance of our studies of differential operators for the understanding of finite data problems via the graph Laplacian. The findings provide insight into parameter choices made in learning algorithms which are based on weighted adjacency matrices; they also provide the basis for analysis of the consistency of various unsupervised and semi-supervised learning algorithms, in the large data limit.

Bayesian inference is used extensively to infer and to quantify the uncertainty in a field of interest from a measurement of a related field when the two are linked by a physical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and/or have prior distributions that are difficult to represent mathematically. In this manuscript we consider the use of Generative Adversarial Networks (GANs) in addressing these challenges. A GAN is a type of deep neural network equipped with the ability to learn the distribution implied by multiple samples of a given field. Once trained on these samples, the generator component of a GAN maps the iid components of a low-dimensional latent vector to an approximation of the distribution of the field of interest. In this work we demonstrate how this approximate distribution may be used as a prior in a Bayesian update, and how it addresses the challenges associated with characterizing complex prior distributions and the large dimension of the inferred field. We demonstrate the efficacy of this approach by applying it to the problem of inferring and quantifying uncertainty in the initial temperature field in a heat conduction problem from a noisy measurement of the temperature at later time.

We consider the problem of estimating the $2D$ vector displacement field in a heterogeneous elastic solid deforming under plane stress conditions. The problem is motivated by applications in quasistatic elastography. From precise and accurate measurements of one component of the $2D$ vector displacement field and very limited information of the second component, the method reconstructs the second component quite accurately. No a priori knowledge of the heterogeneous distribution of material properties is required. This method relies on using a special form of the momentum equations to filter ultrasound displacement measurements to produce more precise estimates. We verify the method with applications to simulated displacement data. We validate the method with applications to displacement data measured from a tissue mimicking phantom, and in-vivo data; significant improvements are noticed in the filtered displacements recovered from all the tests. In verification studies, error in lateral displacement estimates decreased from about $50\%$ to about $2\%$, and strain error decreased from more than $250\%$ to below $2\%$., Comment: 38 pages, 26 figures