Elliptic PDE learning is provably data-efficient.

Partial differential equations (PDE) learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of three-dimensional uniformly elliptic PDEs from input-output data and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success.

A global synchronization theorem for oscillators on a random graph.

Consider n identical Kuramoto oscillators on a random graph. Specifically, consider Erdos-Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability 0 ≤ p ≤ 1. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for p above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, p ∼ log ⁡ ( n ) / n for n ≫ 1. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if p ≫ log ⁡ ( n ) / n, then Erdos-Rényi networks of Kuramoto oscillators are globally synchronizing with high probability as n → ∞. Here, we improve that result by showing that p ≫ log ⁡ ( n ) / n suffices. Our estimates are explicit: for example, we can say that there is more than a 99.9996 % chance that a random network with n = 10 and p > 0.011 17 is globally synchronizing.

Data-driven discovery of Green's functions with human-understandable deep learning.

There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-driven approach for creating a human-machine partnership to accelerate scientific discovery. By collecting physical system responses under excitations drawn from a Gaussian process, we train rational neural networks to learn Green's functions of hidden linear partial differential equations. These functions reveal human-understandable properties and features, such as linear conservation laws and symmetries, along with shock and singularity locations, boundary effects, and dominant modes. We illustrate the technique on several examples and capture a range of physics, including advection-diffusion, viscous shocks, and Stokes flow in a lid-driven cavity.

Sufficiently dense Kuramoto networks are globally synchronizing.

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least µ(n-1) other oscillators. There is a critical value of the connectivity, µc, such that whenever µ>µc, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when µ<µc, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be µc=0.75. In 2020, Lu and Steinerberger proved that µc≤0.7889, and Yoneda, Tatsukawa, and Teramae proved in 2021 that µc>0.6838. This paper proves that µc≤0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.

Dense networks that do not synchronize and sparse ones that do.

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least µ(n-1) other oscillators. Then, there is a critical value of µ above which the system is guaranteed to converge to the in-phase synchronous state for almost all initial conditions. The precise value of µ remains unknown. In 2018, Ling, Xu, and Bandeira proved that if each oscillator is coupled to at least 79.29% of all the others, global synchrony is ensured. In 2019, Lu and Steinerberger improved this bound to 78.89%. Here, we find clues that the critical connectivity may be exactly 75%. Our methods yield a slight improvement on the best known lower bound on the critical connectivity from 68.18% to 68.28%. We also consider the opposite end of the connectivity spectrum, where the networks are sparse rather than dense. In this regime, we ask how few edges one needs to add to a ring of n oscillators to turn it into a globally synchronizing network. We prove a partial result: all the twisted states in a ring of size n=2m can be destabilized by adding just O(nlog2⁡n) edges. To finish the proof, one needs to rule out all other candidate attractors. We have done this for n≤8 but the problem remains open for larger n. Thus, even for systems as simple as Kuramoto oscillators, much remains to be learned about dense networks that do not globally synchronize and sparse ones that do.

Chebyshev Approximation and the Global Geometry of Model Predictions.

Complex nonlinear models are typically ill conditioned or sloppy; their predictions are significantly affected by only a small subset of parameter combinations, and parameters are difficult to reconstruct from model behavior. Despite forming an important universality class and arising frequently in practice when performing a nonlinear fit to data, formal and systematic explanations of sloppiness are lacking. By unifying geometric interpretations of sloppiness with Chebyshev approximation theory, we rigorously explain sloppiness as a consequence of model smoothness. Our approach results in universal bounds on model predictions for classes of smooth models, capturing global geometric features that are intrinsic to their model manifolds, and characterizing a universality class of models. We illustrate this universality using three models from disparate fields (physics, chemistry, biology): exponential curves, reaction rates from an enzyme-catalyzed chemical reaction, and an epidemiology model of an infected population.

Continuous analogues of matrix factorizations.

Analogues of singular value decomposition (SVD), QR, LU and Cholesky factorizations are presented for problems in which the usual discrete matrix is replaced by a 'quasimatrix', continuous in one dimension, or a 'cmatrix', continuous in both dimensions. Two challenges arise: the generalization of the notions of triangular structure and row and column pivoting to continuous variables (required in all cases except the SVD, and far from obvious), and the convergence of the infinite series that define the cmatrix factorizations. Our generalizations of triangularity and pivoting are based on a new notion of a 'triangular quasimatrix'. Concerning convergence of the series, we prove theorems asserting convergence provided the functions involved are sufficiently smooth.