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An overview on deep learning-based approximation methods for partial differential equations

  • * Corresponding author: Benno Kuckuck

    * Corresponding author: Benno Kuckuck

The second author is supported by DFG grant HU1889/7-1

Abstract Full Text(HTML) Figure(1) / Table(9) Related Papers Cited by
  • It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

    Mathematics Subject Classification: Primary: 65-02, 68T07; Secondary: 35-02, 65M99.

    Citation:

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  • Figure 1.  Graphical illustration of a fully-connected feedforward artificial neural network consisting of $ L\in \mathbb N $ affine linear transformations (i.e., consisting of $ L+1 $ layers: one input layer, $ L-1 $ hidden layers, and one output layer) with $ l_0\in \mathbb N $ neurons on the input layer, with $ l_1\in \mathbb N $ neurons on the first hidden layer, with $ l_2\in \mathbb N $ neurons on the second hidden layer, $ \dots $, with $ l_{L-1} $ neurons on the $ (L-1) $-th hidden layer, and with $ l_L $ neurons in the output layer

    Table 1.  The hyperparameters used in the training for the first and third simulation using the deep Galerkin method (see the results in Tables 3 and 5)

    $d$ Batch size Hidden layers Neurons per hidden layer Steps Initial learning rate Learning rate decay
    $1$ $256$ $3$ $41$ $350$ $0.05$ $0.98$
    $2$ $256$ $3$ $42$ $350$ $0.05$ $0.98$
    $5$ $256$ $3$ $45$ $350$ $0.05$ $0.98$
    $10$ $256$ $3$ $50$ $350$ $0.05$ $0.98$
    $20$ $256$ $3$ $60$ $500$ $0.025$ $0.99$
    $50$ $256$ $3$ $90$ $750$ $0.01$ $0.995$
    $100$ $256$ $3$ $140$ $750$ $0.01$ $0.995$
    $200$ $256$ $3$ $240$ $750$ $0.005$ $0.995$
     | Show Table
    DownLoad: CSV

    Table 2.  The hyperparameters used in the training for the second simulation using the deep Galerkin method (see the results in Table 4)

    $d$ Batch size Hidden layers Neurons per hidden layer Steps Initial learning rate Learning rate decay
    $1$ $256$ $3$ $41$ $500$ $0.05$ $0.99$
    $2$ $256$ $3$ $42$ $500$ $0.05$ $0.99$
    $5$ $256$ $3$ $45$ $500$ $0.05$ $0.99$
    $10$ $256$ $3$ $50$ $750$ $0.025$ $0.995$
    $20$ $256$ $3$ $60$ $750$ $0.025$ $0.995$
    $50$ $256$ $3$ $90$ $1000$ $0.01$ $0.998$
    $100$ $256$ $3$ $140$ $1000$ $0.01$ $0.998$
    $200$ $256$ $3$ $240$ $1000$ $0.005$ $0.998$
     | Show Table
    DownLoad: CSV

    Table 3.  Approximations for $u(1/2, 0, 0, \dots, 0)$ where $u$ is the solution of the PDE in (35) with the initial value $ \mathbb R^d\ni x\mapsto u(0, x) = \sqrt{1+||x||^2}\in \mathbb R$ for $d\in\{1, 2, 5, 10, 20, 50, 100, 200\}$ using the deep Galerkin method. For the hyperparameters used in the training, see Table 1. For the Python source code used to obtain these results, see Section 8.2

    $d$ Mean Standard deviation Reference value Absolute $L^1$-error Relative $L^1$-error Average runtime
    $1$ $1.788665$ $0.013821$ $1.836708$ $0.048043$ $0.026157$ $2.48$
    $2$ $2.048171$ $0.025434$ $2.109239$ $0.061302$ $0.029064$ $3.47$
    $5$ $2.684484$ $0.071031$ $2.648632$ $0.057190$ $0.021592$ $6.76$
    $10$ $3.171959$ $0.027321$ $3.208623$ $0.039949$ $0.012451$ $17.84$
    $20$ $4.041396$ $0.112243$ $4.107034$ $0.088848$ $0.021633$ $34.38$
    $50$ $7.193619$ $0.120224$ $7.490964$ $0.297344$ $0.039694$ $133.41$
    $100$ $9.979369$ $0.409066$ $9.808476$ $0.372600$ $0.037988$ $292.10$
    $200$ $13.430278$ $0.247504$ $14.604635$ $1.174357$ $0.080410$ $673.06$
     | Show Table
    DownLoad: CSV

    Table 4.  Approximations for $u(1/2, 0, 0, \dots, 0)$ where $u$ is the solution of the PDE in (35) with the initial value $ \mathbb R^d\ni x\mapsto u(0, x) = 2/(4+||x||^2)\in \mathbb R$ for $d\in\{1, 2, 5, 10, 20, 50, 100, 200\}$ using the deep Galerkin method. For the hyperparameters used in the training, see Table 2. For the Python source code used to obtain these results, see Section 8.2

    $d$ Mean Standard deviation Reference value Absolute $L^1$-error Relative $L^1$-error Average runtime
    $1$ $0.680122$ $0.003512$ $0.677511$ $0.003552$ $0.005243$ $3.62$
    $2$ $0.590964$ $0.009817$ $0.584510$ $0.010066$ $0.017221$ $4.87$
    $5$ $0.374912$ $0.033546$ $0.404080$ $0.030547$ $0.075597$ $9.65$
    $10$ $0.281780$ $0.014043$ $0.258967$ $0.025007$ $0.096566$ $26.19$
    $20$ $0.139796$ $0.013930$ $0.147140$ $0.013351$ $0.090739$ $49.88$
    $50$ $0.076818$ $0.005612$ $0.063228$ $0.013589$ $0.214926$ $175.57$
    $100$ $0.043529$ $0.008632$ $0.032302$ $0.011817$ $0.365843$ $384.41$
    $200$ $0.020501$ $0.014760$ $0.016318$ $0.012173$ $0.745950$ $895.37$
     | Show Table
    DownLoad: CSV

    Table 5.  Approximations for $u(1/2, 0, 0, \dots, 0)$ where $u$ is the solution of the PDE in (35) with the initial value $ \mathbb R^d\ni x\mapsto u(0, x) = \arctan\bigl(\tfrac{||x||}2\bigr)\in \mathbb R$ for $d\in\{1, 2, 5, 10, 20, 50, 100, 200\}$ using the deep Galerkin method. For the hyperparameters used in the training of the DNNs, see Table 1. For the Python source code used to obtain these results, see Section 8.2

    $d$ Mean Standard deviation Reference value Absolute $L^1$-error Relative $L^1$-error Average runtime
    $1$ $0.629165$ $0.029052$ $0.569925$ $0.061155$ $0.107303$ $2.47$
    $2$ $0.900790$ $0.016736$ $0.835350$ $0.065440$ $0.078338$ $3.52$
    $5$ $1.249522$ $0.020660$ $1.201798$ $0.047724$ $0.039711$ $6.77$
    $10$ $1.395483$ $0.022569$ $1.440293$ $0.044975$ $0.031226$ $18.29$
    $20$ $1.593871$ $0.029176$ $1.622937$ $0.035450$ $0.021843$ $34.28$
    $50$ $1.789839$ $0.032926$ $1.785912$ $0.027953$ $0.015652$ $131.69$
    $100$ $1.851764$ $0.034147$ $1.866269$ $0.030308$ $0.016240$ $291.67$
    $200$ $1.626139$ $0.206072$ $1.921878$ $0.295739$ $0.153880$ $678.89$
     | Show Table
    DownLoad: CSV

    Table 6.  The hyperparameters used in the training for the simulations using the deep splitting method (see the results in Tables 79)

    $d$ Batch size Hidden layers Neurons per hidden layer Steps Initial learning rate Learning rate decay
    $1$ $256$ $2$ $51$ $350$ $0.2$ $0.985$
    $2$ $256$ $2$ $52$ $350$ $0.2$ $0.985$
    $5$ $256$ $2$ $55$ $350$ $0.2$ $0.985$
    $10$ $256$ $2$ $60$ $350$ $0.2$ $0.985$
    $20$ $256$ $2$ $70$ $500$ $0.2$ $0.99$
    $50$ $256$ $2$ $100$ $500$ $0.2$ $0.99$
    $100$ $256$ $2$ $150$ $750$ $0.2$ $0.995$
    $200$ $256$ $2$ $250$ $1000$ $0.2$ $0.995$
    $500$ $256$ $2$ $550$ $1000$ $0.2$ $0.995$
    $1000$ $256$ $2$ $1050$ $1250$ $0.2$ $0.998$
     | Show Table
    DownLoad: CSV

    Table 7.  Approximations for $u(1/2, 0, 0, \dots, 0)$ where $u$ is the solution of the PDE in (35) with the initial value $ \mathbb R^d\ni x\mapsto u(0, x) = \sqrt{1+||x||^2}\in \mathbb R$ for $d\in\{1, 2, 5, 10, 20, 50, 100, 200, 500, 1000\}$ using the deep splitting method. For the hyperparameters used in the training of the DNNs, see Table 6. For the Python source code used to obtain these results, see Section 8.3

    $d$ Mean Standard deviation Reference value Absolute $L^1$-error Relative $L^1$-error Average runtime
    $1$ $1.830958$ $0.015773$ $1.836708$ $0.013452$ $0.007324$ $20.22$
    $2$ $2.108470$ $0.014779$ $2.109239$ $0.012182$ $0.005776$ $19.90$
    $5$ $2.642265$ $0.015791$ $2.648632$ $0.014881$ $0.005618$ $20.13$
    $10$ $3.196520$ $0.016279$ $3.208623$ $0.016151$ $0.005034$ $20.31$
    $20$ $4.082424$ $0.019874$ $4.107034$ $0.026197$ $0.006379$ $28.89$
    $50$ $7.480304$ $0.027910$ $7.490964$ $0.022157$ $0.002958$ $29.73$
    $100$ $9.804550$ $0.009534$ $9.808476$ $0.008295$ $0.000846$ $46.51$
    $200$ $14.602816$ $0.036716$ $14.604635$ $0.020845$ $0.001427$ $64.86$
    $500$ $21.386109$ $1.045499$ $22.230142$ $0.844033$ $0.037968$ $75.23$
    $1000$ $31.748045$ $0.126012$ $31.751259$ $0.068201$ $0.002148$ $117.77$
     | Show Table
    DownLoad: CSV

    Table 8.  Approximations for $u(1/2, 0, 0, \dots, 0)$ where $u$ is the solution of the PDE in (35) with the initial value $ \mathbb R^d\ni x\mapsto u(0, x) = 2/(4+||x||^2)\in \mathbb R$ for $d\in\{1, 2, 5, 10, 20, 50, 100, 200, 500, 1000\}$ using the deep splitting method. For the hyperparameters used in the training of the DNNs, see Table 6. For the Python source code used to obtain these results, see Section 8.3

    $d$ Mean Standard deviation Reference value Absolute $L^1$-error Relative $L^1$-error Average runtime
    $1$ $0.678383$ $0.004441$ $0.677511$ $0.003652$ $0.005390$ $20.24$
    $2$ $0.583808$ $0.005164$ $0.584510$ $0.004229$ $0.007235$ $19.93$
    $5$ $0.405637$ $0.003722$ $0.404080$ $0.003680$ $0.009107$ $20.08$
    $10$ $0.257737$ $0.001830$ $0.258967$ $0.001949$ $0.007524$ $20.33$
    $20$ $0.146415$ $0.000674$ $0.147140$ $0.000859$ $0.005841$ $29.08$
    $50$ $0.062905$ $0.000134$ $0.063228$ $0.000323$ $0.005114$ $30.05$
    $100$ $0.032181$ $0.000150$ $0.032302$ $0.000146$ $0.004514$ $46.48$
    $200$ $0.016254$ $0.000023$ $0.016318$ $0.000064$ $0.003941$ $64.93$
    $500$ $0.006542$ $0.000005$ $0.006568$ $0.000026$ $0.003909$ $75.37$
    $1000$ $0.003277$ $0.000010$ $0.003291$ $0.000015$ $0.004467$ $118.31$
     | Show Table
    DownLoad: CSV

    Table 9.  Approximations for $u(1/2, 0, 0, \dots, 0)$ where $u$ is the solution of the PDE in (35) with the initial value $ \mathbb R^d\ni x\mapsto u(0, x) = \arctan\bigl(\tfrac{||x||}2\bigr)\in \mathbb R$ for $d\in\{1, 2, 5, 10, 20, 50, 100, 200, 500, 1000\}$ using the deep splitting method. For the hyperparameters used in the training of the DNNs, see Table 6. For the Python source code used to obtain these results, see Section 8.3

    $d$ Mean Standard deviation Reference value Absolute $L^1$-error Relative $L^1$-error Average runtime
    $1$ $0.561503$ $0.012954$ $0.569925$ $0.012631$ $0.022162$ $20.16$
    $2$ $0.830821$ $0.011572$ $0.835350$ $0.009811$ $0.011745$ $20.19$
    $5$ $1.195350$ $0.007251$ $1.201798$ $0.008002$ $0.006658$ $20.09$
    $10$ $1.436600$ $0.003500$ $1.440293$ $0.004196$ $0.002913$ $20.22$
    $20$ $1.622351$ $0.001493$ $1.622937$ $0.001305$ $0.000804$ $29.17$
    $50$ $1.785940$ $0.000271$ $1.785912$ $0.000214$ $0.000120$ $29.81$
    $100$ $1.866438$ $0.000438$ $1.866269$ $0.000337$ $0.000181$ $46.65$
    $200$ $1.922286$ $0.000091$ $1.921878$ $0.000408$ $0.000212$ $65.01$
    $500$ $1.970859$ $0.000031$ $1.970294$ $0.000566$ $0.000287$ $75.16$
    $1000$ $1.994976$ $0.000086$ $1.994330$ $0.000645$ $0.000324$ $118.21$
     | Show Table
    DownLoad: CSV
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