In particular, we shed light on the popular eikonal loss used for imposing a signed distance function constraint in INR. We show analytically that as the representation power of the network increases, the optimization approaches a partial differential equation (PDE) in the continuum limit that is unstable. We show that this instability can manifest in existing network optimization, leading to irregularities in the reconstructed surface and/or convergence to sub-optimal local minima, and thus fails to capture fine geometric and topological structure. We show analytically how other terms added to the loss, currently used in the literature for other purposes, can actually eliminate these instabilities.

In particular, we shed light on the popular eikonal loss used for imposing a signed distance function constraint in INR. We show analytically that as the representation power of the network increases, the optimization approaches a partial differential equation (PDE) in the continuum limit that is unstable. We show that this instability can manifest in existing network optimization, leading to irregularities in the reconstructed surface and/or convergence to sub-optimal local minima, and thus fails to capture fine geometric and topological structure. We show analytically how other terms added to the loss, currently used in the literature for other purposes, can actually eliminate these instabilities. Based on a similar PDE theory for the continuum limit, we introduce a new regularization term that still counteracts the eikonal instability but without over-regularizing.

As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $\varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $\varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.

This paper studies the $p$-biharmonic equation on graphs, which arises in point cloud processing and can be interpreted as a natural extension of the graph $p$-Laplacian from the perspective of hypergraph. The asymptotic behavior of the solution is investigated when the random geometric graph is considered and the number of data points goes to infinity. We show that the continuum limit is an appropriately weighted $p$-biharmonic equation with homogeneous Neumann boundary conditions. The result relies on the uniform $L^p$ estimates for solutions and gradients of nonlocal and graph Poisson equations. The $L^\infty$ estimates of solutions are also obtained as a byproduct.

In this paper we give a broad overview of the intersection of partial differential equations (PDEs) and graph-based semi-supervised learning. The overview is focused on a large body of recent work on PDE continuum limits of graph-based learning, which have been used to prove well-posedness of semi-supervised learning algorithms in the large data limit. We highlight some interesting research directions revolving around consistency of graph-based semi-supervised learning, and present some new results on the consistency of p-Laplacian semi-supervised learning using the stochastic tug-of-war game interpretation of the p-Laplacian. We also present the results of some numerical experiments that illustrate our results and suggest directions for future work.

Lattice gauge-equivariant convolutional neural networks (L-CNNs) can be used to form arbitrarily shaped Wilson loops and can approximate any gauge-covariant or gauge-invariant function on the lattice. Here we use L-CNNs to describe fixed point (FP) actions which are based on renormalization group transformations. FP actions are classically perfect, i.e., they have no lattice artifacts on classical gauge-field configurations satisfying the equations of motion, and therefore possess scale invariant instanton solutions. FP actions are tree-level Symanzik-improved to all orders in the lattice spacing and can produce physical predictions with very small lattice artifacts even on coarse lattices. We find that L-CNNs are much more accurate at parametrizing the FP action compared to older approaches. They may therefore provide a way to circumvent critical slowing down and topological freezing towards the continuum limit.

In recent decades, science and engineering have been revolutionized by a momentous growth in the amount of available data. However, despite the unprecedented ease with which data are now collected and stored, labeling data by supplementing each feature with an informative tag remains to be challenging. Illustrative tasks where the labeling process requires expert knowledge or is tedious and time-consuming include labeling X-rays with a diagnosis, protein sequences with a protein type, texts by their topic, tweets by their sentiment, or videos by their genre. In these and numerous other examples, only a few features may be manually labeled due to cost and time constraints. How can we best propagate label information from a small number of expensive labeled features to a vast number of unlabeled ones? This is the question addressed by semi-supervised learning (SSL). This article overviews recent foundational developments on graph-based Bayesian SSL, a probabilistic framework for label propagation using similarities between features. SSL is an active research area and a thorough review of the extant literature is beyond the scope of this article. Our focus will be on topics drawn from our own research that illustrate the wide range of mathematical tools and ideas that underlie the rigorous study of the statistical accuracy and computational efficiency of graph-based Bayesian SSL.

The goal of this paper is to determine the laws of observed trajectories assuming that there is a mechanical system in the background and using these laws to continue the observed motion in a plausible way. The laws are represented by neural networks with a limited number of parameters. The training of the networks follows the Extreme Learning Machine idea. We determine laws for different levels of embedding, thus we can represent not only the equation of motion but also the symmetries of different kinds. In the recursive numerical evolution of the system, we require the fulfillment of all the observed laws, within the determined numerical precision. In this way, we can successfully reconstruct both integrable and chaotic motions, as we demonstrate in the example of the gravity pendulum and the double pendulum.

Tackling semi-supervised learning problems with graph-based methods have become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, e.g., of differential operators. A popular strategy here is $p$-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled data. For $p<\infty$ continuum limits of this approach were studied using tools from $\Gamma$-convergence. For the case $p=\infty$, which is referred to as Lipschitz learning, continuum limits of the related infinity-Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using $\Gamma$-convergence. In particular, we define a sequence of functionals which approximate the largest local Lipschitz constant of a graph function and prove $\Gamma$-convergence in the $L^\infty$-topology to the supremum norm of the gradient as the graph becomes denser. Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the Hausdorff distance. We apply our results to nonlinear ground states and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.

We consider the problem of recovering a function input of a differential equation formulated on an unknown domain $M$. We assume to have access to a discrete domain $M_n=\{x_1, \dots, x_n\} \subset M$, and to noisy measurements of the output solution at $p\le n$ of those points. We introduce a graph-based Bayesian inverse problem, and show that the graph-posterior measures over functions in $M_n$ converge, in the large $n$ limit, to a posterior over functions in $M$ that solves a Bayesian inverse problem with known domain. The proofs rely on the variational formulation of the Bayesian update, and on a new topology for the study of convergence of measures over functions on point clouds to a measure over functions on the continuum. Our framework, techniques, and results may serve to lay the foundations of robust uncertainty quantification of graph-based tasks in machine learning. The ideas are presented in the concrete setting of recovering the initial condition of the heat equation on an unknown manifold.