GNGTS 2023 - Atti del 41° Convegno Nazionale

Session 3.3 ______ ___ GNGTS 2023 Machine Learning To Approximate The Objective Function In A Global Inversion Of Ert Data F. Macelloni, M. Aleardi, E. Stucchi Department of Earth Sciences, University of Pisa, Italy Introduction An inverse problem aims to determine the parameters of a model from the measured data. During the inversion, the parameters are iteratively updated and the associated data are calculated from the resulting models through the forward modelling operation. This last step is usually the most time consuming of the process. In this work we aim to reduce this computational effort by replacing the forward modelling and the data misfit calculation with a properly trained Neural Network (NN) model that approximates the objective function. NNs, the basis of Machine Learning (ML) algorithms, consist of computing models that can learn the underlying relationship between their inputs and their outputs. We decide to apply the ML-based inversion to Electrical Resistivity Tomography (ERT), a nonlinear and ill-posed inverse problem, usually solved through deterministic gradient-based methods that linearize the problem around an initial solution (Aleardi et al. , 2021). We tackle the problem with a global optimization method, Genetic Algorithms (GA): this allows a good exploration of the solution space treating the models as individuals of populations and performing on them operations inspired by the natural selection processes. In this way, and compared to local approaches, GAs reduce the risk of entrapment in a local minimum of the objective function. Besides the problem of a forward model computationally expensive, there is another issue that makes a global optimization ERT inversion computationally challenging: the large dimensional parameter space and the associated curse of dimensionality problem. To this end we employ the Discrete Cosine Transform (DCT) to reduce the number of unknowns to invert for. This technique is similar to the Fourier Transform, but it uses only cosines as bases function to signal reconstruction so that the computed coefficients are real numbers. Therefore, the DCT of a signal (expressing, in this case, the resistivity model) indicates the signal’s energy distribution in the frequency domain. Since most of the signal’s energy is usually expressed by low-order DCT coefficients, retaining only these ones we can use this technique for model compression. In the inverse problem, the selected coefficients become the unknown parameters to retrieve (Vinciguerra et al. , 2021). By reducing the dimensionality of the problem, the computational effort required to deal with it also decreases.