Achieving High Accuracy in Neural Network Training for PDEs with Energy Natural Gradient Descent.
Dienstag, 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.
The fundamental gap conjecture
Dienstag, 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.