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Elastic continuum mechanical models are widely used to compute deformations due to pressure changes in buried cavities, such as magma reservoirs. In general, analytical models are fast but can be inaccurate as they do not correctly satisfy boundary conditions for many geometries, while numerical models are slow and may require specialized expertise and software. To overcome these limitations, we trained supervised machine learning emulators (model surrogates) based on parallel partial Gaussian processes which predict the output of a finite element numerical model with high fidelity but >1,000× greater computational efficiency. The emulators are based on generalized nondimensional forms of governing equations for finite non‐dipping spheroidal cavities in elastic halfspaces. Either cavity volume change or uniform pressure change boundary conditions can be specified, and the models predict both surface displacements and cavity (pore) compressibility. Because of their computational efficiency, using the emulators as numerical model surrogates can greatly accelerate data inversion algorithms such as those employing Bayesian Markov chain Monte Carlo sampling. The emulators also permit a comprehensive evaluation of how displacements and cavity compressibility vary with geometry and material properties, revealing the limitations of analytical models. Our open‐source emulator code can be utilized without finite element software, is suitable for a wide range of cavity geometries and depths, includes an estimate of uncertainties associated with emulation, and can be used to train new emulators for different source geometries.
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PARALLEL PARTIAL EMULATION IN APPLICATIONS
Inference, optimization, and inverse problems are but three examples of mathematical operations that require the repeated solution of a complex system of mathematical equations. To this end, surrogates are often used to approximate the output of these large computer simulations, providing fast and cheap approximation solutions. Statistical emulators are surrogates that, in addition to predicting the mean behavior of the system, provide an estimate of the error in that prediction. Classical Gaussian stochastic process emulators predict scalar outputs based on a modest number of input parameters. Making predictions across a space-time field of input variables is not feasible using classical Gaussian process methods. Parallel partial emulation is a new statistical emulator methodology that predicts a field of outputs based on the input parameters. Parallel partial emulation is constructed as a Gaussian process in parameter space, but no correlation among space or time points is assumed. Thus the computational work of parallel partial emulation scales as the cube of the number of input parameters (as traditional Gaussian Process emulation) and linearly with a space-time grid. The numerical methods used in numerical simulations are often designed to exploit properties of the equations tobe solved. For example, modern solvers for hyperbolic conservation laws satisfy conservation at each time step, insuring overall conservation of the physical variables. Similarly, symplectic methods are used to solve Hamiltonian problems in physics. It is of interest, then, to study whether parallel partial emulation predictions inherit properties possessed by the simulation outputs. Does an emulated solution of a conservation law preserve the conserved quantities? Does an emulator of a Hamiltonian system preserve the energy? This paper investigates the properties of emulator predictions, in the context of systems of partial differential equations. We study conservation properties for three different kinds ofequations-conservation laws, reaction-diffusion systems, and a Hamiltonian system.We also investigate the effective convergence, in parameter space, of the predicted solution of a highly nonlinear system modeling shape memory alloys.
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- Award ID(s):
- 2053874
- PAR ID:
- 10556486
- Publisher / Repository:
- Begel House
- Date Published:
- Journal Name:
- International Journal for Uncertainty Quantification
- Volume:
- 14
- Issue:
- 6
- ISSN:
- 2152-5080
- Page Range / eLocation ID:
- 1 to 15
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1615/Int.J.UncertaintyQuantification.2024048538
