Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method
Abstract
1. Introduction
2. Approximating Schemes for Reflected PDEs
2.1. Nonlinear Parabolic Reflected PDEs
2.2. From Reflected PDEs to Related Reflected BSDEs
2.3. Discretizing via Two Approaches
3. Numerical Experiments
3.1. Deep C-N Algorithm for Solving High-Dimensional Nonlinear Reflected PDEs
- ( and the same settings in 2 to 4) is a forward iterative procedure, which is determined by approximating scheme (6); this procedure does not contain any parameters that need to be optimized.
- is a forward iterative procedure too, which is characterized by approximating scheme (7). As in the previous step, no parameters need to be optimized in this operation.
- is the key step in the whole calculating procedure. Our goal in this step is approximating the spatial gradients, and meanwhile, the weights are optimized in the (N − 1) sub-networks.
- is a forward iteration procedure that yields the neural network’s final output as the unique approximation of , totally characterized by approximating scheme (14).
3.2. Allen–Cahn Equation
3.3. American Options
4. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Number of Iteration Step | Standard | Relative -Approximate Error | Relative -Approximate Error | Mean Value of Loss Function | Standard Deviation of Loss Function | |
---|---|---|---|---|---|---|
0 | 0.4740 | 0.0514 | 7.9775 | 0.9734 | 0.11630 | 0.02953 |
1000 | 0.1446 | 0.0340 | 1.7384 | 0.6436 | 0.00550 | 0.00344 |
2000 | 0.0598 | 0.0058 | 0.1318 | 0.1103 | 0.00029 | 0.00006 |
3000 | 0.0530 | 0.0002 | 0.0050 | 0.0041 | 0.00023 | 0.00001 |
4000 | 0.0528 | 0.0002 | 0.0030 | 0.0022 | 0.00020 | 0.00001 |
Number of Iteration Step | Standard | Relative -Approximate Error | Relative -Approximate Error | Mean Value of Loss Function | Standard Deviation of Loss Function | |
---|---|---|---|---|---|---|
0 | 0.5021 | 0.0791 | 0.2979 | 0.449313 | 0.137191 | 0.043493 |
2000 | 0.0659 | 0.0083 | 0.0131 | 0.011521 | 0.000407 | 0.000142 |
4000 | 0.0569 | 0.0021 | 0.0002 | 0.000040 | 0.000201 | 0.000027 |
6000 | 0.0531 | 0.0002 | 0.0002 | 0.000013 | 0.000118 | 0.000240 |
8000 | 0.0529 | 0.0002 | 0.0002 | 0.000156 | 0.000055 | 0.000012 |
10,000 | 0.0528 | 0.0001 | 0.0001 | 0.000117 | 0.000030 | 0.000010 |
Dimensions | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 |
---|---|---|---|---|---|---|---|---|
Deep BSDE | 3.415 × 10−4 | 1.886 × 10−4 | 1.44 × 10−4 | 1.029 × 10−4 | 9.973 × 10−5 | 5.330 × 10−5 | 7.789 × 10−5 | 5.774 × 10−5 |
Deep C-N | 1.095 × 10−4 | 3.095 × 10−5 | 1.853 × 10−5 | 1.435 × 10−5 | 1.398 × 10−5 | 1.303 × 10−5 | 1.337 × 10−5 | 1.506 × 10−5 |
Models | Dimensions | Value | Reference | Relative Error |
---|---|---|---|---|
Deep C-N | 5 | 0.10720 | 0.10738 | 0.17% |
RDBDP | 5 | 0.10657 | 0.10738 | 0.75% |
Deep BSDE | 5 | NC | 0.10738 | NC |
Deep C-N | 10 | 0.12687 | 0.12996 | 2.38% |
RDBDP | 10 | 0.12829 | 0.12996 | 1.29% |
Deep BSDE | 10 | NC | 0.12996 | NC |
Deep C-N | 20 | 0.15140 | 0.15100 | 0.27% |
RDBDP | 20 | 0.14430 | 0.15100 | 4.38% |
Deep BSDE | 20 | NC | 0.15100 | NC |
Deep C-N | 40 | 0.16213 | 0.16800 | 3.49% |
RDBDP | 40 | 0.16167 | 0.16800 | 3.77% |
Deep BSDE | 40 | NC | 0.16800 | NC |
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Shi, X.; Zhang, X.; Tang, R.; Yang, J. Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method. Math. Comput. Appl. 2023, 28, 79. https://doi.org/10.3390/mca28040079
Shi X, Zhang X, Tang R, Yang J. Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method. Mathematical and Computational Applications. 2023; 28(4):79. https://doi.org/10.3390/mca28040079
Chicago/Turabian StyleShi, Xiaowen, Xiangyu Zhang, Renwu Tang, and Juan Yang. 2023. "Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method" Mathematical and Computational Applications 28, no. 4: 79. https://doi.org/10.3390/mca28040079
APA StyleShi, X., Zhang, X., Tang, R., & Yang, J. (2023). Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method. Mathematical and Computational Applications, 28(4), 79. https://doi.org/10.3390/mca28040079