Scientific machine learning is a burgeoning discipline for mixing machine learning into scientific simulation. Use cases of this field include automated discovery of physical equations and accelerating physical simulators. However, making the analyses of this field automated will require building a set of tools that handle stiff and ill-conditioned models without requiring user tuning. The purpose of this talk is to demonstrate how the methods and tools of scientific machine learning can be consolidated to give a single high performance and robust software stack. We will start by describing universal differential equations, a flexible mathematical object which is able to represent methodologies for equation discovery, 100-dimensional differential equation solvers, and discretizations of physics-informed neural networks. Then we will showcase how adjoint sensitivity analysis on the universal differential equation solving process gives rise to efficient and stiffly robust training methodologies for a large variety of scientific machine learning problems. With this understanding of differentiable programming we will describe how the Julia SciML Software Organization is utilizing this foundation to provide high performance tools for deploying battery powered airplanes, improving the energy efficiency of buildings, allow for navigation via the Earth's magnetic field, and more.

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Whitney elements on simplices are perhaps the most widely used finite elements in computational electromagnetics. They offer the simplest construction of polynomial discrete differential forms on simplicial complexes. Their associated degrees of freedom (dofs) have a very clear physical meaning and give a recipe for discretizing physical balance laws, e.g., Maxwell’s equations. As interest grew for the use of high order schemes, such as hp-finite element or spectral element methods, higher-order extensions of Whitney forms have become an important computational tool, appreciated for their better convergence and accuracy properties. However, it has remained unclear what kind of cochains such elements should be associated with: Can the corresponding dofs be assigned to precise geometrical elements of the mesh, just as, for instance, a degree of freedom for the space of Whitney 1-forms belongs to a specific edge? We address this localization issue. Why is this an issue? The existing constructions of high order extensions of Whitney elements follow the traditional FEM path of using higher and higher “moments” to define the needed dofs. As a result, such high order finite k-elements in d dimensions include dofs associated to q-simplices, with k < q ≤ d, whose physical interpretation is obscure. The present paper offers an approach based on the so-called “small simplices”, a set of subsimplices obtained by homothetic contractions of the original mesh simplices, centered at mesh nodes (or more generally, when going up in degree, at points of the principal lattice of each original simplex). Degrees of freedom of the high-order Whitney k-forms are then associated with small simplices of dimension k only.  We provide an explicit  basis for these elements on simplices and we justify this approach from a geometric point of view (in the spirit of Hassler Whitney's approach, still successful 30 years after his death).   

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The Spin(7) and SU(4) structures on a Calabi-Yau 4-fold give rise to certain first order PDEs defining special Yang-Mills connections: the Spin(7) instanton equations and the Hermitian Yang-Mills (HYM) equations respectively. The latter are stronger than the former. In 1998 C. Lewis proved that -over a compact base space- the existence of an HYM connection implies the converse. In this talk we demonstrate that the equivalence of the two gauge-theoretic problems fails to hold in generality. We do this by studying the invariant solutions on a highly symmetric noncompact Calabi-Yau 4-fold: the Stenzel manifold. We give a complete description of the moduli space of irreducible invariant Spin(7) instantons with structure group SO(3) on this space and find that the HYM connections are properly embedded in it. This moduli space reveals an explicit example of a sequence of Spin(7) instantons bubbling off near a Cayley submanifold. The missing limit is an HYM connection, revealing a potential relationship between the two equation systems.

In the talk I will survey the fast growing field of metric measure spaces satisfying a lower bound on Ricci Curvature, in a synthetic sense via optimal transport. Particular emphasis will be given to discuss how such (possibly non-smooth) spaces naturally (and usefully) extend the class of smooth Riemannian manifolds with Ricci curvature bounded below.

I discuss a ``running vacuum cosmological model'' of a string-inspired
Universe, in which gravitational anomalies play an important role, in
inducing, through condensates of primordial gravitational waves, an early de
Sitter inflationary phase, during which constant (in cosmic time)
backgrounds of the antisymmetric (Kalb-Ramond (KR)) tensor field of the
massless bosonic string multiplet remain undiluted until the exit from
inflation and well into the subsequent radiation era. During the radiation
phase, such backgrounds, which violate spontaneously Lorentz and CPT
symmetry, induce lepton asymmetry (Leptogenesis) in models involving
right-handed neutrinos. Chiral matter is generated in the model at the exit
phase of inflation, and this leads to the cancellation of gravitational
anomalies in the post inflationary universe. During the radiation era, non
perturbative effects can also be held responsible for the generation of a

potential for the gravitational axion, associated in (3+1)-dimensions with
the field strength of the KR field, which can thus play the role of a Dark
Matter component. In the talk, I discuss the underlying formalism and argue
in favour of the consistency of a theory with gravitational anomalies in the
early Universe. I connect the energy density of such a universe with that of
the so called ``running-vacuum model'' in which the vacuum energy density is
expressed in terms of even powers of the Hubble parameter, which in general
depends on cosmic time. The gravitational-wave condensate induces a term in
the energy density  proportional to the fourth-power of the Hubble parameter
H^4 , which is responsible for the early de Sitter phase, during which the
Hubble parameter is approximately a constant. I also discuss briefly a
connection of this string inspired model with the Swampland and weak gravity
conjectures and explain how consistency with such conjectures is achieved,
despite the fact that the model is compatible with slow-roll inflationary
phenomenology.

Tissue folding during animal development involves an intricate interplay
of cell shape changes, cell division, cell migration, cell
intercalation, and cell differentiation that obfuscates the underlying
mechanical principles. However, a simpler instance of tissue folding
arises in the green alga Volvox: its spherical embryos turn themselves
inside out at the close of their development. This inversion arises from
cell shape changes only.

In this talk, I will present a model of tissue folding in which these
cell shape changes appear as variations of the intrinsic stretches and
curvatures of an elastic shell. I will show how this model reproduces
Volvox inversion quantitatively, explains mechanically the arrest of
inversion observed in mutants, and reveals the spatio-temporal
regulation of different biological driving processes. I will close with
two examples illustrating the challenges of nonlinearity in tissue
folding: (i) constitutive nonlinearity leading to nonlocal elasticity in
the continuum limit of discrete cell sheet models; (ii) geometric
nonlinearity in large bending deformations of morphoelastic shells.
 

We briefly review mathematical models of viscous deformable interfaces (such as plasma membranes) leading to fluid equations posed on (evolving) 2D surfaces embedded in $R^3$. We further report on some recent advances in understanding and numerical simulation of the resulting fluid systems using an unfitted finite element method.

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Reconstructing continuous signals based on a small number of discrete samples is a fundamental problem across science and engineering. In practice, we are often interested in signals with ``simple'' Fourier structure -- e.g., those involving frequencies within a bounded range, a small number of frequencies, or a few blocks of frequencies. More broadly, any prior knowledge about a signal's Fourier power spectrum can constrain its complexity.  Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct.

We formalize this intuition by showing that, roughly speaking, a continuous signal from a given class can be approximately reconstructed using a number of samples equal to the statistical dimension of the allowed power spectrum of that class. We prove that, in nearly all settings, this natural measure tightly characterizes the sample complexity of signal reconstruction.

Surprisingly, we also show that, up to logarithmic factors, a universal non-uniform sampling strategy can achieve this optimal complexity for any class of signals. We present a simple, efficient, and general algorithm for recovering a signal from the samples taken. For bandlimited and sparse signals, our method matches the state-of-the-art. At the same time, it gives the first computationally and sample efficient solution to a broad range of problems, including multiband signal reconstruction and common kriging and Gaussian process regression tasks.

Our work is based on a novel connection between randomized linear algebra and the problem of reconstructing signals with constrained Fourier structure. We extend tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in the signal fitting problem. We believe that these extensions are of independent interest and serve as a foundation for tackling a broad range of continuous time problems using randomized methods.

This is joint work with Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker and Amir Zandieh

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