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Data-driven evolutions of critical points

  • * Corresponding author

    * Corresponding author 
S.A. acknowledges the support of the DFG Transregio 109 "Discretization in Geometry and Dynamics". M.F. acknowledges the support of the DFG Project "Identification of Energies from Observation of Evolutions" and the DFG SPP 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization". R.H. acknowledges the support of the DFG-FWF IGDK1754 "Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures" and the hospitality of TUM during the preparation of this work
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  • In this paper we are concerned with the learnability of energies from data obtained by observing time evolutions of their critical points starting at random initial equilibria. As a byproduct of our theoretical framework we introduce the novel concept of mean-field limit of critical point evolutions and of their energy balance as a new form of transport. We formulate the energy learning as a variational problem, minimizing the discrepancy of energy competitors from fulfilling the equilibrium condition along any trajectory of critical points originated at random initial equilibria. By $ \Gamma $-convergence arguments we prove the convergence of minimal solutions obtained from finite number of observations to the exact energy in a suitable sense. The abstract framework is actually fully constructive and numerically implementable. Hence, the approximation of the energy from a finite number of observations of past evolutions allows one to simulate further evolutions, which are fully data-driven. As we aim at a precise quantitative analysis, and to provide concrete examples of tractable solutions, we present analytic and numerical results on the reconstruction of an elastic energy for a one-dimensional model of thin nonlinear-elastic rod.

    Mathematics Subject Classification: Primary: 49N80, 68T05; Secondary: 34A55, 70F17.


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  • Figure 1.  Reconstructions $ \hat a $ (left) and $ \hat a' $ (right) in dotted red and true $ a $ and $ a' $ in blue. The underlying red circles depict the (adaptive) nodes in $ \Lambda $. Below you see the distribution $ \tilde \eta $ of available data

    Figure 2.  Reconstruction of $ a' $ with increasing number $ {N_e} $ of measurements. The true $ a' $ is depicted by the blue curve while the red line depicts the reconstruction $ \hat a' $ following (5.23). The red circles at the bottom of the figures depict the position of nodes of the underlying mesh $ \Lambda_N $. The histogram below describes the density of the available information encoded by the probability measure $ \tilde{\eta} $ for the case $ N_e = 60 $

    Figure 3.  Reconstruction of $ a' $ for varying $ N $ and $ D(N) = 4N $ and distribution $ \tilde \eta $ for $ N = 60 $. Graphic as described in Figure 2

    Figure 4.  Reconstruction of $ a' $ for varying $ N $ and fixed $ D(N) = 300 $ and distribution $ \tilde \eta $. Graphics as described in Figure 2

    Figure 5.  Impact of different constants $ M_2 $ for constraints on $ a'' $ with $ \|a''\| \leq [2,5;20,1000] $ on reconstructs are depicted, showing improved approximations for increasing $ M_2 $. Graphics as in Figure 2

    Figure 6.  Impact of changing constraint $ M_1 $ bounding the values of $ a' $, showing a projection-like behaviour of the reconstruction for small $ M_1 $. Graphics as in Figure 2

    Figure 7.  Left showing $ \hat x_\varepsilon(t) $ (dark dashed) and $ x_\varepsilon(t) $ (line bright) with data from Section 5.2.2 at times $ t = [0.2 $, $ 0.4 $, $ 0.6 $, $ 0.8 $, $ 1.0] $, right with data from Section 5.2.3

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