Weak Convergence Effects of Vector Fields Constrained by General Linear Partial Differential Equations

The numerical resolution of continuum models in science requires approximating the infinite-dimensional solution space with finite-dimensional surrogates, implemented using finitely many measurements. A cornerstone step in the theoretical validation of such models is proving that approximate surrogates converge to idealized solutions assuming infinitely resolved measurements. The main difficulty lies in the subtle interaction between weak convergence, the mode of convergence mimicking physical/statistical measurements, and the nonlinearity of the governing partial differential equations (PDE). This project studies weak convergence effects of solutions to general linear PDE systems, identifying weakly continuous or lower semicontinuous nonlinear quantities. The project is thus an extension of the Compensated Compactness theory. As the structure of linear PDE is governed by Algebraic Geometry, establishing the main results requires blending this field with Real, Complex, and Fourier Analysis, yielding deep theoretical results. On the practical side, numerical investigations draw from the algebraic structure to create a complete feedback loop with the theory: experiments inform proofs and proofs validate computational algorithms. The implementation uses modern Artificial Intelligence (AI) algorithms, in the form of physics-constrained neural networks and Gaussian processes, leveraging advanced GPU-accelerated computing to improve on state-of-the-art numerical methods. Furthermore, the project provides research training opportunities for undergraduate and graduate students, as well as postdoctoral researchers, who will be thoroughly prepared to enter the AI-related workforce at either industrial or academic levels. The main goal of this project is to establish the equivalence between A-quasiconvexity and weak lower semicontinuity of integral functionals defined on A-free sequences, where A is a linear vectorial PDE operator (system). The relevance of this achievement lies in the difficulty and complexity of establishing the claim in full generality. There are few contributions to this problem in the last three decades, aside from the case of so-called operators of constant rank. The first part of the project establishes nontrivial estimates for linear PDE systems, at the interface of Algebraic Geometry and Algebraic Analysis, expanding on the celebrated Ehrenpreis Palamodov theorem. The second part of the project concerns the interaction between A-quasiconvex functionals and the estimates on A-free sequences obtained in the first part, by developing a new approximation scheme. The nonlinear results in the second part are also interpreted in the language of generalized Young measures, a highly efficient tool in describing weak convergence effects, which sits at the interface between Measure Theory and Functional Analysis. The third part covers the practical implementation of generative models that define probability measures on the solution space of a given linear PDE system. This involves defining priors inspired by the Ehrenpreis Palamodov theorem that solve the PDE exactly, thereby enabling regression algorithms or efficient training of neural networks. As a by-product of the third part, this project consolidates AI as a driving force behind advances in Scientific Computing. The project facilitates continuous enrollment of undergraduate students in a research program training them in the implementation of AI methods for computing solutions of general linear PDE. Graduate students and postdoctoral researchers receive dual training in both theory and practice, making them attractive for jobs in both science and industry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2555785 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Bogdan Raita | Institution: Georgetown University, WASHINGTON, DC | Award Amount: $201,757 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2555785 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2555785.html