UC Santa Barbara

In this dissertation, we propose a series of machine learning strategies to address numerical difficulties in the level-set method. First, we introduce a few data-driven solutions to improve mean-curvature estimations along smooth interfaces in two- and three-dimensional free boundary problems. To this end, we have leveraged level-set, gradient, and curvature information to train error-correcting neural networks. These models have evolved from a preliminary network-only approach and its subsequent extension into our first hybrid inference system. However, our most recent curvature multilayer perceptrons are part of a more sophisticated machine-learning-enhanced solver. This solver uses the numerical mean-curvature approximations as a starting point. Then, our models quantify and yield curvature corrective terms on demand based on the interfacial degree of under-resolution.

Our second contribution is a two-dimensional machine-learning-augmented semi-Lagrangian scheme. Its goal is to improve mass preservation. To build this hybrid scheme, we have resorted to image super-resolution methodologies. In particular, our passive-transport solver features an error-quantifying multilayer perceptron. Such a network produces on-the-fly corrections for coarse-grid trajectories to mimic the interface motion in much finer grids. In this research, we have found that these advection models operate better with input vectors that incorporate not only level-set but also other contextual information, such as gradient, curvature, velocity, and positional data. Also critical in this case is an alternating mechanism with the standard semi-Lagrangian scheme. Together, these advection systems can conserve area better by smoothing the moving front and counteracting the undermining effects of numerical viscosity.

Finally, we demonstrate through several experiments that our machine-learning-enhanced solvers can outperform the standard schemes for under-resolved and steep interface regions. Likewise, we show these solvers can attain the same accuracy as the conventional frameworks at higher grid resolutions but require only a fraction of the cost. Further, our data-driven systems work locally with interface stencils, operate transparently on uniform and adaptive grids, and can be readily integrated into existing codebases. Thus, this dissertation confirms that machine learning provides feasible mechanisms for neutralizing mass loss by improving geometrical calculations in low resolutions.