A Deep Learning Approach for the Computation of Curvature in the Level-Set Method

We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from circular interfaces immersed in uniform grids of various resolutions. These multilayer perceptrons process the level-set values from mesh points next to the free boundary and output the dimensionless curvature at their closest locations on the interface. Accuracy analyses involving irregular interfaces, both in uniform and adaptive grids, show that our models are competitive with traditional numerical schemes in the $L^1$ and $L^2$ norms. In particular, our neural networks approximate curvature with comparable precision in coarse resolutions, when the interface features steep curvature regions, and when the number of iterations to reinitialize the level-set function is small. Although the conventional numerical approach is more robust than our framework, our results have unveiled the potential of machine learning for dealing with computational tasks where the level-set method is known to experience difficulties. We also establish that an application-dependent map of local resolutions to neural models can be devised to estimate mean curvature more effectively than a universal neural network.Comment: Submitted to SIAM Journal on Scientific Computin

A Hybrid Inference System for Improved Curvature Estimation in the Level-Set Method Using Machine Learning

We present a novel hybrid strategy based on machine learning to improve curvature estimation in the level-set method. The proposed inference system couples enhanced neural networks with standard numerical schemes to compute curvature more accurately. The core of our hybrid framework is a switching mechanism that relies on well established numerical techniques to gauge curvature. If the curvature magnitude is larger than a resolution-dependent threshold, it uses a neural network to yield a better approximation. Our networks are multilayer perceptrons fitted to synthetic data sets composed of sinusoidal- and circular-interface samples at various configurations. To reduce data set size and training complexity, we leverage the problem's characteristic symmetry and build our models on just half of the curvature spectrum. These savings lead to a powerful inference system able to outperform any of its numerical or neural component alone. Experiments with static, smooth interfaces show that our hybrid solver is notably superior to conventional numerical methods in coarse grids and along steep interface regions. Compared to prior research, we have observed outstanding gains in precision after training the regression model with data pairs from more than a single interface type and transforming data with specialized input preprocessing. In particular, our findings confirm that machine learning is a promising venue for reducing or removing mass loss in the level-set method.Comment: Submitte

Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method

We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, here we develop a pair of models in $\mathbb{R}^3$, regardless of the mesh size. Our feedforward networks ingest transformed level-set, gradient, and curvature data to fix numerical mean-curvature approximations selectively for interface nodes. To reduce the problem's complexity, we have used the Gaussian curvature to classify stencils and fit our models separately to non-saddle and saddle patterns. Non-saddle stencils are easier to handle because they exhibit a curvature error distribution characterized by monotonicity and symmetry. While the latter has allowed us to train only on half the mean-curvature spectrum, the former has helped us blend the data-driven and the baseline estimations seamlessly near flat regions. On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent information beyond what is known. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches. Our approach to building their data sets is systematic but gleans samples randomly while ensuring well-balancedness. We have also resorted to standardization and dimensionality reduction and integrated regularization to minimize outliers. In addition, we leverage curvature rotation/reflection invariance to improve precision at inference time. Several experiments confirm that our proposed system can yield more accurate mean-curvature estimations than modern particle-based interface reconstruction and level-set schemes around under-resolved regions

Transformative machine learning algorithms for free boundary problems

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

Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-Set Method

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-C\'ardenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. The core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost

Compilación de Proyectos de Investigacion de 1984-2002

Instituto Politecnico Nacional. UPIICS