September  2020, 2(3): 257-278. doi: 10.3934/fods.2020012

Spectral methods to study the robustness of residual neural networks with infinite layers

IGPM, RWTH Aachen University, Germany

* Corresponding author: Torsten Trimborn

Published  August 2020

Fund Project: This research is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2023 Internet of Production - 390621612 and supported also by DFG HE5386/18, 19 and DFG 320021702/GRK2326. T. Trimborn acknowledge the support by the ERS Prep Fund - Simulation and Data Science

Recently, neural networks (NN) with an infinite number of layers have been introduced. Especially for these very large NN the training procedure is very expensive. Hence, there is interest to study their robustness with respect to input data to avoid unnecessarily retraining the network.

Typically, model-based statistical inference methods, e.g. Bayesian neural networks, are used to quantify uncertainties. Here, we consider a special class of residual neural networks and we study the case, when the number of layers can be arbitrarily large. Then, kinetic theory allows to interpret the network as a dynamical system, described by a partial differential equation. We study the robustness of the mean-field neural network with respect to perturbations in initial data by applying UQ approaches on the loss functions.

Citation: Torsten Trimborn, Stephan Gerster, Giuseppe Visconti. Spectral methods to study the robustness of residual neural networks with infinite layers. Foundations of Data Science, 2020, 2 (3) : 257-278. doi: 10.3934/fods.2020012
References:
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show all references

References:
[1]

R. Abgrall and S. Mishra, Uncertainty quantification for hyperbolic systems of conservation laws, Handbook of numerical methods for hyperbolic problems, Handb. Numer. Anal., Elsevier/North-Holland, Amsterdam, 18 (2017), 507–544.  Google Scholar

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[4]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, 1st edition, Progress in nonlinear differential equations and their applications, Birkhäuser, Switzerland, 2016. doi: 10.1007/978-3-319-32062-5.  Google Scholar

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[12]

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[13]

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[14]

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[15]

I. CraveroG. PuppoM. Semplice and G. Visconti, CWENO: Uniformly accurate reconstructions for balance laws, Math. Comp., 87 (2018), 1689-1719.  doi: 10.1090/mcom/3273.  Google Scholar

[16]

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[17]

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Y. Deng, Y. Shen, K. Chen and H. Jin, Training recurrent neural network through moment matching for nlp applications, (2018), 3353–3357. Google Scholar

[19]

B. Després, G. Poëtte and D. Lucor, Robust uncertainty propagation in systems of conservation laws with the entropy closure method, Uncertainty Quantification in Computational Fluid Dynamics, Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 92 (2013), 105–149. doi: 10.1007/978-3-319-00885-1_3.  Google Scholar

[20]

D. Funaro, Polynomial Approximation of Differential Equations, Lecture Notes in Physics. New Series m: Monographs, 8. Springer-Verlag, Berlin, 1992.  Google Scholar

[21]

S. Gerster and M. Herty, Entropies and symmetrization of hyperbolic stochastic Galerkin formulations, Communications in Computational Physics, 27 (2020), 639-671.   Google Scholar

[22]

S. GersterM. Herty and A. Sikstel, Hyperbolic stochastic Galerkin formulation for the $p$-system, J. Comput. Phys., 395 (2019), 186-204.  doi: 10.1016/j.jcp.2019.05.049.  Google Scholar

[23]

S. Gerster, M. Herty and H. Yu, Hypocoercivity of stochastic Galerkin formulations for stabilization of kinetic equations, RWTH, preprint, 1–21. Google Scholar

[24]

F. S. Gharehchopogh, Neural networks application in software cost estimation: A case study, in 2011 International Symposium on Innovations in Intelligent Systems and Applications, IEEE, (2011), 69–73. doi: 10.1109/INISTA.2011.5946160.  Google Scholar

[25]

M. B. Giles, Multilevel Monte Carlo methods, Springer Proceedings in Mathematics & Statistics, Monte Carlo and Quasi-Monte Carlo Methods, 65 (2012), 83–103. doi: 10.1007/978-3-642-41095-6_4.  Google Scholar

[26]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  Google Scholar

[27]

F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lect. Notes Appl. Math. Mech., Springer, 3 (2016), 1–144. doi: 10.1007/978-3-319-26883-5_1.  Google Scholar

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Figure 1.  Illustration of a neural network with $ L+1 = 5 $ total layers
Figure 2.  Illustration of the SimResNet with $ L+1 = 5 $ total layers
Figure 3.  Top left panel: Regression of the target function $ h(x) = \tanh(x) $ on $ \mathbb{H} = [-4,4] $ provided by the deep artificial neural networks with the statistical quantities computed with the Monte-Carlo method and the stochastic collocation method with noise level $ \varepsilon = 0.3 $. Top right panel: Expected value of the quantity of interest. Bottom left panel: Variance of the quantity of interest. Bottom right panel: Convergence rate of the Monte-Carlo method
Figure 4.  Top left panel: classification problem with the statistical quantities computed with the Monte-Carlo method and the stochastic collocation method with noise level $ \varepsilon = 0.5 $. Top right panel: expected value of the quantity of interest. Bottom left panel: variance of the quantity of interest. Bottom right panel: convergence rate of the Monte-Carlo method
Figure 5.  CASE 1: Projection of initial data and root mean squared error (RMSE) with exactly computed mean and variance. Upper panel: The left $ y $-axis shows the gPC modes $ \hat{g}_k(0,x) := \big\langle g_0(x,\cdot ) \, \phi_k( \cdot ) \big\rangle_\rho $ of initial values. The mean $ \mathbb{E}\big[ g_0({x},\boldsymbol{\eta}) \big] = \hat{g}_0(0,x) $ is plotted in red, the others in green. The confidence region, which contains $ \mathbb{P} $-a.s. all realisations $ g_0\big(x,\boldsymbol{\eta}(\omega)\big) $ is gray shaded. The right $ y $-axis shows the variance corresponding to different gPC truncations. The upper left panel shows for each fixed signal $ x $ the root mean squared error (RMSE) of the gPC truncations. The right panel shows the total RMSE for various gPC truncations
Figure 6.  CASE 2: Projection of initial data and root mean squared error (RMSE) with exactly computed mean and variance. Upper panel: The blue lines describe deviations from the mean of initial data (red). Two realisations are shown in black, which are in the confidence region (gray shaded). The lower left panel shows for each fixed signal $ x $ the root mean squared error (RMSE) of the gPC truncations. The right panel shows the total RMSE for various gPC truncations
Figure 7.  CASE 1: Solutions to the stochastic Galerkin formulation with truncation $ K = 30 $ and discretization $ \Delta x = 0.01 $. The upper panels show in red the first gPC mode $ \hat{g}_0(t,x) $, which describes the temporal evolution of the mean. The other modes are plotted in green, which determine the spread of the stochastic perturbations. The lower panels show the probability distributions of the random quantity $ g(t,x,\boldsymbol{\eta}) $ for a fixed signal $ x\in\{2,8\} $ at time $ t = 0 $ (left), $ t = 0.5 $ (middle) and $ t = 1 $ (right)
Figure 8.  CASE 2: Solutions to the stochastic Galerkin formulation with truncation $ K = 10 $ and discretization $ \Delta x = 0.01 $. The left $ y $-axes of the first and second panel show the mean and deviations. The right $ y $-axes show the gPC modes $ \hat{g}_1,\ldots,\hat{g}_3 $. The right panels show the probability distributions of the random quantity $ g(t,1,\boldsymbol{\eta}) $ at time $ t = 0 $ (left), $ t = 0 $ and $ t = 5 $
Figure 9.  CASE 1: Mean, variance and distribution for random loss. The left panels show for each fixed signal $ x $ the mean and the variance of the random loss. The middle panels show for various noise levels $ \varepsilon $ the mean and variance of the total random loss. The right panel shows the probability distributions of the random loss in $ x = 2 $ and $ x = 8 $ at time $ t = 0.5 $ and $ t = 1 $
Figure 10.  CASE 2: Mean, variance and distribution for random loss. The left panels show for each fixed signal $ x $ the mean and the variance of the random loss. The middle panels show for various noise levels $ \varepsilon $ the mean and variance of the total random loss. The right panels show the probability distributions of the random loss in $ x = 1 $, respectively at time $ t = 0 $ and $ t = 5 $
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