Discrete Exterior Calculus

I will present our ongoing program at Princeton [1–7], which aims to develop a structure-preserving theoretical framework for modeling the dynamics of charge fields coupled to gauge fields. The focus is on capturing physical phenomena beyond the scope of perturbation theory in the radiation–matter coupling, and on enabling accurate derivation and numerical simulation of the resulting equations, both in the quantum regime and relevant semiclassical limits, using simplicial mesh-based numerical methods. Applications will be presented to some problems in scalar quantum electrodynamics, capturing the dynamics of low-lying collective excitations in superconducting quantum devices, and to the massive Schwinger Model, which describes the relativistic dynamics of charged leptons interacting with an electromagnetic gauge field in 1+1 dimensions. Within this framework, a DEC-inspired approach is used to derive an effective discretized field theory that describes the dynamics of spatially and/or temporally coarse-grained fields within a finite space-time volume, tailored to accurately capture the features resolvable by a physical measurement apparatus.

[1] D. Pham et al., “Flux-Based Three-Dimensional Electrodynamic Modeling Approach to Superconducting Circuits and Materials.” Phys. Rev. A 107, 053704 (2023).[2] K. Sinha et al., “Radiative Properties of an Artificial Atom Coupled to a Josephson Junction Array.” Phys. Rev. A 106, 033714 (2022).[3] T. Maldonado et al., “Negative Electrohydrostatic Pressure Between Superconducting Bodies.” Physical Review B 110.1 (2024).[4] W. Fan et al., “Model Order Reduction for Open Quantum Systems Based on Measurement-Adapted Time-Coarse Graining.” arxiv:2410.23116[5] D. Pham et al., “Spectral Theory for Non-Linear Superconducting Microwave Systems: Extracting Relaxation Rates and Mode Hybridization.”[6] D. Pham et al., “Long-Time Soliton Dynamics via a Coarse-Grained Space-Time Method.", arxiv:2504.12286[7] T. Maldonado et al., “Mesoscopic Theory of the Josephson Junction.” Physical Review B 111, L140505 (2025).

We provide a finite element discretization of $ell$-form-valued $k$-forms on triangulation in $mathbb{R}^{n}$ for general $k$, $ell$ and $n$ and any polynomial degree. The construction generalizes finite element Whitney forms for the de~Rham complex and their high-order and distributional versions, the Regge finite elements and the Christiansen--Regge elasticity complex, the TDNNS element for symmetric stress tensors, the MCS element for traceless matrix fields, the Hellan--Herrmann--Johnson (HHJ) elements for biharmonic equations, and discrete divdiv and Hessian complexes in [Hu, Lin, and Zhang, 2025]. The construction discretizes the Bernstein--Gelfand--Gelfand (BGG) diagrams. Applications of the construction include discretization of strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension.

While the standard definition of a finite element is the one given by Ciarlet in his 1978 book, we have been developing an alternative approach, called Finite Element Systems, based on category theory. Then the mesh is interpreted as a category, and the discrete function spaces attached to cells, provide a contravariant functor to the category of finite dimensional vector spaces. In other words a finite element is a particular kind of presheaf. Restrictions from cells in the mesh to subcells play a crucial role. The unisolvence of degrees of freedom in Ciarlets definition corresponds to a softness condition on the sheaf. Sheaf theory provides a useful setting in which to compare continuous and discrete theories. Indeed the continuous theory can also be described with sheaves of Banach spaces. The language works well for sequences such as differential forms and elasticity, and unifying de Rham theorems on cohomology have been proved in this setting.

We present a framework for combining expressive transformer architectures from computer vision with conventional Whitney form constructions to identify structure preserving reduced order models from data. Much like how conditional generative models allow sampling of images from a text prompt, this allows sampling of FEEC models from sensor measurements, which we use to construct digital twins that can both predict and assimilate in real time. We then introduce a recent extension to build hybridizable neural integrators; while significant effort focuses on using vision transformers to build auto-regressive forecasters of physical systems, our framework allows recasting of the problem in terms of a trainable mixed FEEC space in time. Analysis highlights geometric conservation structure, energy stability, and uniform boundedness of parameter sensitivities independent of the number of rollouts, resolving a current major issue in long term stability for AI-driven forecasting tools.