Christopher Earls

Robert Stephany, Christopher Earls

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, $U$, to approximate the system response function and a sparse, trainable vector, $ξ$, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of $U$ and $ξ$ using a loss function that (1) makes $U$ approximate the system response function, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $ξ$ characterizes, and (3) promotes sparsity in $ξ$ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.

Andrew Loeb, Christopher Earls

The simulation of heat flow through heterogeneous material is important for the design of structural and electronic components. Classical analytical solutions to the heat equation PDE are not known for many such domains, even those having simple geometries. The finite element method can provide approximations to a weak form continuum solution, with increasing accuracy as the number of degrees of freedom in the model increases. This comes at a cost of increased memory usage and computation time; even when taking advantage of sparse matrix techniques for the finite element system matrix. We summarize recent approaches in solving problems in structural mechanics and steady state heat conduction which do not require the explicit assembly of any system matrices, and adapt them to a method for solving the time-depended flow of heat. These approaches are highly parallelizable, and can be performed on graphical processing units (GPUs). Furthermore, they lend themselves to the simulation of heterogeneous material, with a minimum of added complexity. We present the mathematical framework of assembly-free FEM approaches, through which we summarize the benefits of GPU computation. We discuss our implementation using the OpenCL computing framework, and show how it is further adapted for use on multiple GPUs. We compare the performance of single and dual GPUs implementations of our method with previous GPU computing strategies from the literature and a CPU sparse matrix approach. The utility of the novel method is demonstrated through the solution of a real-world coefficient inverse problem that requires thousands of transient heat flow simulations, each of which involves solving a 1 million degree of freedom linear system over hundreds of time steps.

Robert Stephany, Christopher Earls

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms to train a neural network, $U$, to approximate the PDE solution while simultaneously identifying the governing PDE. This approach yields an algorithm that is robust to noise and can discover a range of PDEs directly from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.

Robert Stephany, Christopher Earls

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are sparse and noisy. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a principled sparse regression algorithm to identify the hidden dynamics that govern a system's response. The first network learns the system response function, while the second learns a hidden PDE describing the system's evolution. We then use a parameter-free sparse regression algorithm to extract a human-readable form of the hidden PDE from the second network. We implement our approach in an open-source library called PDE-READ. Our approach successfully identifies the governing PDE in six benchmark examples. We demonstrate that our approach is robust to both sparsity and noise and it, therefore, holds promise for application to real-world observational data.

Raphaël Sarfati, Haley Moller, Toni J. B. Liu et al.

Large language models use high-dimensional latent spaces to encode and process textual information. Much work has investigated how the conceptual content of words translates into geometrical relationships between their vector representations. Fewer studies analyze how the cumulative information of an entire prompt becomes condensed into individual embeddings under the action of transformer layers. We use literary pieces to show that information about intangible, rather than factual, aspects of the prompt are contained in deep representations. We observe that short excerpts (10 - 100 tokens) from different novels separate in the latent space independently from what next-token prediction they converge towards. Ensembles from books from the same authors are much more entangled than across authors, suggesting that embeddings encode stylistic features. This geometry of style may have applications for authorship attribution and literary analysis, but most importantly reveals the sophistication of information processing and compression accomplished by language models.

Harshwardhan Praveen, Jacob Brown, Christopher Earls

In this work, we present a mesh-independent, data-driven library, chebgreen, to mathematically model one-dimensional systems, possessing an associated control parameter, and whose governing partial differential equation is unknown. The proposed method learns an Empirical Green's Function for the associated, but hidden, boundary value problem, in the form of a Rational Neural Network from which we subsequently construct a bivariate representation in a Chebyshev basis. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library, expressed as points on a manifold of Quasimatrices, while the associated singular values are interpolated with Lagrange polynomials.