Computing curvature for volume of fluid methods using machine learning

In spite of considerable progress, computing curvature in Volume of Fluid (VOF) methods continues to be a challenge. The goal is to develop a function or a subroutine that returns the curvature in computational cells containing an interface separating two immiscible fluids, given the volume fraction in the cell and the adjacent cells. Currently, the most accurate approach is to fit a curve (2D), or a surface (3D), matching the volume fractions and finding the curvature by differentiation. Here, a different approach is examined. A synthetic data set, relating curvature to volume fractions, is generated using well- defined shapes where the curvature and volume fractions are easily found and then machine learning is used to fit the data (training). The resulting function is used to find the curvature for shapes not used for the training and implemented into a code to track moving interfaces. The results suggest that using machine learning to generate the relationship is a viable approach that results in reasonably accurate predictions

Comparison of methods for curvature estimation from volume fractions

This paper evaluates and compares the accuracy and robustness of curvature estimation methods for three-dimensional interfaces represented implicitly by discrete volume fractions on a Cartesian mesh. The height function (HF) method is compared to three paraboloid fitting methods: fitting to the piecewise linear interface reconstruction centroids (PC), fitting to the piecewise linear interface reconstruction volumetrically (PV), and volumetrically fitting (VF) the paraboloid directly to the volume fraction field. The numerical studies presented in this work find that while the curvature error from the VF method converges with second-order accuracy as with the HF method for static interfaces represented by exact volume fractions, the PV method best balances low curvature errors with low computational cost for dynamic interfaces when the interface reconstruction and advection are coupled to a two-phase Navier-Stokes solver

Deep learning of interfacial curvature: a symmetry-preserving approach for the volume of fluid method

Estimation of interface curvature in surface-tension dominated flows is a remaining challenge in Volume of Fluid (VOF) methods. Data-driven methods are recently emerging as a promising alternative in this domain. They outperform conventional methods on coarser grids but diverge with grid refinement. Furthermore, unlike conventional methods, data-driven methods are sensitive to coordinate system and sign conventions, thus often fail to capture basic symmetry patterns in interfaces. The present work proposes a new data-driven strategy which conserves the symmetries in a cost-effective way and delivers consistent results over a wide range of grids. The method is based on artificial neural networks with deep multilayer perceptron (MLP) architecture which read volume fraction fields on regular grids. The anti-symmetries are preserved with no additional cost by employing a neural network model with input normalization, odd-symmetric activation functions and bias-free neurons. The symmetries are further conserved by height-function inspired rotations and averaging over several different orientations. The new symmetry-preserving MLP model is implemented into a flow solver (OpenFOAM) and tested against conventional schemes in the literature. It shows superior performance compared to its standard counterpart and has similar accuracy and convergence properties with the state-of-the-art conventional method despite using smaller stencil.Comment: Preprint under review in Journal of Computational Physic

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

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

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

Modélisation dynamique inverse de tissus - Apprentissage profond à l'aide de simulations basées sur la physique

Inverse problems arise in various physical domains and solving them from real-world visual observations poses a significant challenge due to the high dimensional nature of the data. Furthermore gathering enough observations that a data driven model can accurately capture the complete distribution of a physical phenomenon is often intractable. In this work we use deep learning to solve inverse problems by applying two basic principles. Deep learning models can be trained using synthetic data generated from physics based simulations. And the employed simulator itself needs to be verified for physical accuracy thus allowing the model to learn the exact physical phenomenon that is desired.To validate the simulator, we introduce rich and compact physical protocols, originally proposed in soft matter physics literature to measure physical parameters. These protocols can be easily replicated in a simulator to test the physical correctness of the model, and the validity of the simulator.We solve the inverse measurement problem of estimating contact friction in soft-bodies which otherwise requires a specialized physics bench and entails tedious acquisition protocols. This makes the prospect of a purely non-invasive, video-based measurement technique particularly attractive. Previous works have shown that such a video-based estimation is feasible for material parameters using deep learning, but this has never been applied to the friction estimation problem which results in even more subtle visual variations. Since acquiring a large dataset for this problem is impractical, we generate it using a frictional contact simulator. As the simulator has been calibrated and verified using controlled experiments, the results are not only visually plausible, but physically-correct enough to match observations made at the macroscopic scale. We propose to our knowledge the first non-invasive measurement network and adjoining synthetic training dataset for estimating cloth friction at contact, for both cloth-hard body and cloth-cloth contacts. We also acquire an extensive dataset of real world experiments for testing. Both the training and test datasets have been made freely available to the community.We also utilize the same protocol for solving the inverse measurement problem of estimating the deformed curvature of a suspended Kirchhoff rod. In order to do such estimation on physical rods, we utilize a deep learning model to visually predict a curvature field from a suspended rod. As creating a dataset from physical rods (even if synthetically constructed), that faithfully covers a representative manifold of deformed curvatures is intractable, we rely on generating such a dataset from a verified simulator. Our work shows a promising way forward for utilizing deep learning models as part of an inversion measurement pipeline.Des problèmes inverses surviennent dans divers domaines physiques et les résoudre à partir d'observations visuelles du monde réel pose un défi important en raison de la nature hautement dimensionnelle des données. De plus, rassembler suffisamment d'observations pour qu'un modèle basé sur les données puisse capturer avec précision la distribution complète d'un phénomène physique est souvent insoluble. Dans ce travail, nous utilisons l'apprentissage profond pour résoudre des problèmes inverses en appliquant deux principes de base. Les modèles d'apprentissage profond peuvent être entraînés à l'aide de données synthétiques générées à partir de simulations basées sur la physique. Et la précision physique du simulateur employé, lui-même, doit être vérifiée, permettant ainsi au modèle d'apprendre le phénomène physique exact souhaité.Afin de valider le simulateur, nous introduisons des protocoles physiques riches et compacts, proposés à l'origine dans la littérature de physique de la matière molle pour mesurer des paramètres physiques. Ces protocoles peuvent être facilement répliqués dans un simulateur pour tester l'exactitude physique du modèle et la validité du simulateur.Nous résolvons le problème de mesure inverse de l'estimation du frottement de contact dans les corps mous qui nécessite sinon un banc de physique spécialisé et un protocole d'acquisition fastidieux. Cela rend la perspective d'une technique de mesure purement non invasive basée sur la vidéo particulièrement attrayante. Des travaux antérieurs ont montré qu'une telle estimation basée sur la vidéo est réalisable pour les paramètres de matériaux en utilisant l'apprentissage profond, mais cela n'a jamais été appliqué au problème d'estimation de la friction qui entraîne des variations visuelles encore plus subtiles. Étant donné qu'il n'est pas pratique d'acquérir un grand ensemble de données pour ce problème, nous le générons à l'aide d'un simulateur de contact frictionnel. Comme le simulateur a été calibré et vérifié à l'aide d'expériences contrôlées, les résultats sont non seulement visuellement plausibles, mais suffisamment corrects physiquement pour correspondre aux observations faites à l'échelle macroscopique. Nous proposons à notre connaissance le premier réseau de mesure non invasif et un jeu de données d'entraînement synthétique adjacent pour estimer le frottement du tissu au contact, à la fois pour les contacts tissu-corps dur et tissu-tissu. Nous acquérons également un vaste ensemble de données d'expériences du monde réel pour les tests. Les ensembles de données de formation et de test ont été mis gratuitement à la disposition de la communauté.Nous utilisons également le même protocole pour résoudre le problème de mesure inverse de l'estimation de la courbure déformée d'une tige de Kirchhoff suspendue. Afin de faire une telle estimation sur des tiges physiques, nous utilisons un modèle d'apprentissage profond pour prédire visuellement un champ de courbure à partir d'une tige suspendue. Comme la création d'un ensemble de données à partir de tiges physiques (même si elles sont synthétiquement construites), qui couvre fidèlement une variété représentative de courbures déformées est insoluble, nous comptons sur la génération d'un tel ensemble de données à partir d'un simulateur vérifié. Notre travail montre une voie prometteuse pour l'utilisation de modèles d'apprentissage profond dans le cadre d'un pipeline de mesure d'inversion