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1. A tutorial on the Bayesian statistical approach to inverse problems

   https://doi.org/10.1063/5.0154773  

   Waqar, Faaiq_G; Patel, Swati; Simon, Cory_M (November 2023, APL Machine Learning)

   Inverse problems are ubiquitous in science and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input–output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a “solution” that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our Tutorial aims to reach a wide range of scientists and engineers. 

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2. Novel Tool for Selecting Surrogate Modeling Techniques for Surface Approximation

   https://doi.org/10.1016/B978-0-323-88506-5.50071-1  

   Williams, B. Cremaschi (January 2021, Computer aided chemical engineering)

   Turkay, M. Aydin (Ed.)

   Surrogate models are used to map input data to output data when the actual relationship between the two is unknown or computationally expensive to evaluate for several applications, including surface approximation and surrogate-based optimization. Many techniques have been developed for surrogate modeling; however, a systematic method for selecting suitable techniques for an application remains an open challenge. This work compares the performance of eight surrogate modeling techniques for approximating a surface over a set of simulated data. Using the comparison results, we constructed a Random Forest based tool to recommend the appropriate surrogate modeling technique for a given dataset using attributes calculated only from the available input and output values. The tool identifies the appropriate surrogate modeling techniques for surface approximation with an accuracy of 87% and a precision of 86%. Using the tool for surrogate model form selection enables computational time savings by avoiding expensive trial-and-error selection methods. 

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3. Surrogate Modeling with Gaussian Processes for an Inverse Problem in Polymer Dynamics

   https://doi.org/10.1142/S0219876221430039  

   Chouhan, Pankaj; Shanbhag, Sachin (February 2022, International Journal of Computational Methods)

   When rheological models of polymer blends are used for inverse modeling, they can characterize polymer mixtures from rheological observations. This requires repeated evaluation of potentially expensive rheological models. We explored surrogate models based on Gaussian processes (GP-SM) as a cheaper alternative for describing the rheology of polydisperse binary blends. We used the time-dependent diffusion double reptation (TDD-DR) model as the true model; it takes a 5-dimensional input vector specifying the binary blend as input and yields a function called the relaxation spectrum as output. We used the TDD-DR model to generate training data of different sizes [Formula: see text], via Latin hypercube sampling. The optimal values of the GP-SM hyper-parameters, assuming a separable covariance kernel, were obtained by maximum likelihood estimation. The GP-SM interpolates the training data by design and offers reasonable predictions of relaxation spectra with uncertainty estimates. In general, the accuracy of GP-SMs improves as the size of the training data [Formula: see text] increases, as does the cost for training and prediction. The optimal hyper-parameters were found to be relatively insensitive to [Formula: see text]. Finally, we considered the inverse problem of inferring the structure of the polymer blend from a synthetic dataset generated using the true model. Surprisingly, the solution to the inverse problem obtained using GP-SMs and TDD-DR was qualitatively similar. GP-SMs can be several orders of magnitude cheaper than expensive rheological models, which provides a proof-of-concept validation for using GP-SMs for inverse problems in polymer rheology. 

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4. Solving Bayesian Inverse Problems via Variational Autoencoders

   Goh, H. (January 2021, Proceeding of Machine Learning Research, 2nd Annual Conference on Mathematical and Scientific Machine Learning)

   Joan Bruna, Jan S (Ed.)

   In recent years, the field of machine learning has made phenomenal progress in the pursuit of simulating real-world data generation processes. One notable example of such success is the variational autoencoder (VAE). In this work, with a small shift in perspective, we leverage and adapt VAEs for a different purpose: uncertainty quantification in scientific inverse problems. We introduce UQ-VAE: a flexible, adaptive, hybrid data/model-constrained framework for training neural networks capable of rapid modelling of the posterior distribution representing the unknown parameter of interest. Specifically, from divergence-based variational inference, our framework is derived such that most of the information usually present in scientific inverse problems is fully utilized in the training procedure. Additionally, this framework includes an adjustable hyperparameter that allows selection of the notion of distance between the posterior model and the target distribution. This introduces more flexibility in controlling how optimization directs the learning of the posterior model. Further, this framework possesses an inherent adaptive optimization property that emerges through the learning of the posterior uncertainty. Numerical results for an elliptic PDE-constrained Bayesian inverse problem are provided to verify the proposed framework. 

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5. Optimization framework for patient‐specific modeling under uncertainty

   https://doi.org/10.1002/cnm.3665  

   Mineroff, Joshua; Pokuri, Balaji Sesha Sarath; Ganapathysubramanian, Baskar; Krishnamurthy, Adarsh (December 2022, International Journal for Numerical Methods in Biomedical Engineering)

   Abstract Estimating a patient‐specific computational model's parameters relies on data that is often unreliable and ill‐suited for a deterministic approach. We develop an optimization‐based uncertainty quantification framework for probabilistic model tuning that discovers model inputs distributions that generate target output distributions. Probabilistic sampling is performed using a surrogate model for computational efficiency, and a general distribution parameterization is used to describe each input. The approach is tested on seven patient‐specific modeling examples using CircAdapt, a cardiovascular circulatory model. Six examples are synthetic, aiming to match the output distributions generated using known reference input data distributions, while the seventh example uses real‐world patient data for the output distributions. Our results demonstrate the accurate reproduction of the target output distributions, with a correct recreation of the reference inputs for the six synthetic examples. Our proposed approach is suitable for determining the parameter distributions of patient‐specific models with uncertain data and can be used to gain insights into the sensitivity of the model parameters to the measured data. 

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