Achieving High Accuracy in Neural Network Training for PDEs with Energy Natural Gradient Descent.
Tuesday, 18.4.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We will talk about energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds to the Newton direction modulo an orthogonal projection onto the model's tangent space. We present numerical results illustrating that energy natural gradient descent yields highly accurate solutions with errors several orders of magnitude smaller than what is obtained when training PINNs with standard optimizers like gradient descent, Adam or BFGS, even when those are allowed significantly more computation time.
Informalizing formalized mathematics using the Lean theorem prover
Friday, 21.4.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
One of the applications of interactive theorem provers in\npure mathematics is being able to produce machine-verified formal proofs. I will talk about a less-obvious application, which is using formalized mathematics to author interactive informal expositions. I will demonstrate a prototype of an "auto-informalization" system written in Lean that presents the reader with an interface to view proofs at a desired level of detail. I will also discuss thoughts on the impact of such tools in mathematics. This is joint work with Patrick Massot.\n
Automorphisms of the Boutet de Monvel algebra
Monday, 24.4.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
In a remarkable work, Duistermaat and Singer in 1976 studied the algebras of all classical pseudodifferential operators on smooth (boundaryless) manifolds. They gave a description of order preserving algebra isomorphism between the algebras of classical pseudodifferential operators of two manifolds. The subject of this talk is the generalisation of their results to manifolds with boundary. The role of the algebra of pseudodifferential operators that we are interested in is the Boutet de Monvel algebra.\n\nThe main fact of life about manifold with boundary is that vector fields do not define global flows and the "boundary conditions" are a way of dealing with this problem. The Boutet de Monvel algebra corresponds to the choice of local boundary conditions and is, effectively, a non-commutative completion of the manifold. One can think of it as a parametrised version of the classical Toeplitz algebra as a completion of the half-space.\n\nWhat appears in the study of automorphisms are Fourier integral operators and we will try to explain their appearance - both in boundaryless and boundary case. as it turns out, the non-trivial boundary case introduces both some complications but also some simplifications of the analysis involved, Once this is done, the analysis that we need reduces to a high degree to relatively classical results about automorphisms and homology of the Toeplitz algebra and some basic facts from K-theory.\n\nThis is a joint work in progress with Elmar Schrohe.
The fundamental gap conjecture
Tuesday, 25.4.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
At the end of his study [J. Statist. Phys., 83] on thermodynamic functions\nof a free boson gas, van den Berg conjectured that the difference between the\ntwo smallest eigenvalues\n\nΓ\nV\n(Ω) := λ\nV\n2\n(Ω) − λ\nV\n1\n(Ω);\n\nof the Schr¨odinger operator −∆ + V on a convex domain Ω in R\nd\n, d ≥ 1,\nequipped with homogeneous Dirichlet boundary conditions satisfies\n\nΓ\nV\n(Ω) ≥ Γ (ID) = 3π\n2\nD2\n, (1)\n\nwhere ID is the interval (−D/2, D/2) of length D = diameter(Ω) . The term\nΓ\nV\n(Ω) is called the fundamental gap and describes an important physical quantity: for example, in statistical mechanics, ΓV\n(Ω) measures the energy needed to\njump from the ground state to the next excited eigenstate, or computationally,\nit can control the rate of convergence of numerical methods to compute large\neigenvalue problems [SIAM, 2011]. Thus, one is interested in (optimal) lower\nbounds on ΓV\n(Ω). Since the late 80s, the fundamental conjecture (1) attracts\nconsistently the attention of many researcher including M. S. Ashbaugh & R.\nBenguria [Proc. Amer. Math. Soc., 89], R. Schoen and S.-T. Yau Camb. Press,\n94., B. Andrews and J. Clutterbuck [J. Amer. Math.\nSoc., 11].\n\nIn this talk, I present new results on the fundamental gap conjecture (1)\nfor the Schr¨odinger operator −∆ + V on a convex domain Ω equipped with\nRobin boundary conditions. In particular, we present a proof of this conjecture\nin dimension one, and mention results for the p-Laplacian.\n\nThe talk is based on the joint works [1, 2] with B. Andrews and J. Clutterbuck.\nReferences\n[1] Ben Andrews, Julie Clutterbuck, and Daniel Hauer. Non-concavity of the\nRobin ground state. Camb. J. Math., 8(2):243–310, 2020.\n[2] Ben Andrews, Julie Clutterbuck, and Daniel Hauer. The fundamental gap\nfor a one-dimensional Schr¨odinger operator with Robin boundary conditions.\nProc. Amer. Math. Soc., 149(4): 1481–1493, 2021.
Strong distributivity and games on posets
Tuesday, 25.4.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
A forcing order is said to be \(<\bkappa\)-distributive iff it does not add new sequences of length \(<\bkappa\). A sufficient but not necessary condition for this is that the forcing is \(<\bkappa\)-closed, i.e. any \(<\bkappa\)-sequence of conditions has a lower bound. We introduce a strenghtening of \(<\bkappa\)-distributivity called strong \(<\bkappa\)-distributivity which can replace \(<\bkappa\)-closure in many applications. A main benefit of this property is that a \(<\bkappa\)-closed forcing remains strongly \(<\bkappa\)-distributive in any extension by a \(\bkappa\)-cc. order, even though it no longer necessarily is \(<\bkappa\)-closed.
Syzygies of the cotangent complex
Friday, 28.4.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The cotangent complex is an important but difficult to understand object associated to a map of commutative rings (or schemes). It is connected with some easier to compute invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies inside the cotangent complex. Quillen conjectured that, for maps of finite flat dimension, the cotangent complex can only be bounded for locally complete intersection homomorphisms. This was proven by Avramov in 1999. I will explain how to get a new proof by paying attention to these syzygies, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module.
Combinatorics of toric bundles
Friday, 28.4.23, 14:00-15:00, SR 119
Toric bundles are fibre bundles which have a toric variety as a fiber. One particularly studied class of toric bundles are horospherical varieties which are toric bundles over generalized flag varieties. Similar to toric varieties, toric bundles admit a combinatorial description via polyhedral geometry. In my talk, I will explain such a combinatorial description, and describe a couple of results which rely on it. In particular, I will present a generalization of the BKK theorem and the Fano criterion for toric bundles.