UC Davis

Modern mathematical practice is often informed by computational methods, and conversely, improvements in computational methods can be informed by mathematics. In this thesis we pull on these rich threads and present an interdisciplinary research program which begins with the mathematical analysis of shock formation for the fractal Burgers equation and culminates in applying intuitions gleaned from our research program on fluid mechanics to improve the statistical efficiency of multi-modal large language models.We begin with a problem from pure analysis in Chapter 2: constructing and precisely describing the shock formation process for the fractal Burgers equation. Solving this problem requires extending existing theory to cover fractional dissipation. Our result is the first to successfully deal with the fractional dissipation term through a careful pointwise analysis of the fractional Laplacian.During the completion of this first result, we made frequent use of numerical simulation to inform our analytic methodology. We show that the dynamics of shock formation converge asymptotically to the stable self-similar Burgers dynamics. A thorough analysis of these dynamics is challenging using traditional numerical methods, however recently authors have shown that physics-informed neural networks (PINNS) are well-suited to solving these often unstable numerical problems. In our studies using PINNs we quickly realized that the standard technique of applying autodifferentiation introduces unnecessary repeated computation. In Chapter 3 we connect the PINN training scenario to a classical machine learning algorithm through a novel insight. We show that a simple extension of the TangentProp algorithm can reduce the runtime of PINN training from exponential to quasilinar and we run empirical experiments to demonstrate the efficacy of our theory.From our work on PINNs we developed a novel viewpoint regarding what we call overfitting MLP net- works. We demonstrate the payoff of this perspective by applying overfitting MLPs to a problem from the large-language modeling literature. After our work on PINNs, we began to explore ways in which we could apply our ideas to improving the performance of large language models. We observed that multi-modal vision language models use inefficient priors to ingest images, and that this ingestion process can be stream- lined using the theory and intuitions developed during our study of PINNs. In particular, the overfitting perspective on PINNs frames the learning target as reducing training loss, subject to some auxiliary condition. In the realm MLLMs we show that the standard method for encoding images can be improved by using a content aware selection strategy. However using this selection strategy introduces an interpolation problem which must be solved. This problem cannot be solved satisfactorily using standard bilinear or bicubic interpolation, but it can be phrased as an MLP overfitting problem with an auxiliary condition, exactly like a PINN. In Chapter 4 we are able to leverage our intuitions from studying PINNs and apply them in this use-case as well to train overfit interpolation networks and dramatically improve MLLM inference on a challenging benchmark without having to retrain or fine-tune the underlying LLM. Our solution to this problem demonstrates the advantage of leveraging mathematical understanding to improve the statistical efficiency of a model, rather than relying purely on compute (re-training).