| Citation: |
Shijing Si, Yijin Gao, Haixia Sun, Aifan Ling. SOLVING STOCHASTIC INVENTORY MODEL EQUATION WITH DEEP NEURAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 2819-2840. doi: 10.11948/20250377
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SOLVING STOCHASTIC INVENTORY MODEL EQUATION WITH DEEP NEURAL NETWORKS
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Abstract
The joint optimization of dynamic pricing and inventory control in stochastic inventory systems has been extensively investigated in the previous literature before. A general common approach is to transform the dynamic system into a time-independent ordinary differential equation that represents the stable state. In this study, we extend the model to incorporate the general case with time-dependent dynamics. Consequently, the problem can be reformulated as solving a corresponding partial differential equation (PDE) that lacks an analytical solution. However, assuming an analytical solution encounters certain limitations: (1) Ensuring the accurate existence and uniqueness of the solution is challenging, and (2) completely solving the reduced ordinary differential equation system becomes difficult from theoretical sense. To address these challenges, efficient numerical methods are commonly employed. Nevertheless, for high-dimensional problems, numerical methods like finite element or finite difference methods tend to be computationally slow and suffer from accuracy loss. To overcome these issues, we propose the use of a deep learning technique named Physics-informed Neural Networks (PINN) to solve this complex system, which is the first application of PINN application in this research topic. Simulated examples are presented to validate the usefulness of PINN methods.
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References
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Shijing Si, Yijin Gao, Haixia Sun, Aifan Ling. SOLVING STOCHASTIC INVENTORY MODEL EQUATION WITH DEEP NEURAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 2819-2840. doi: 10.11948/20250377
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Figure 1.
Schematic of a PINN for solving the PDE in Eq. (3.1) with boundary (BC) and initial conditions (IC).
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Figure 2.
Initial value is
$ x $ -
Figure 3.
Initial value is
$ x $ $ u(x, t) $ $ t $ -
Figure 4.
Initial value is
$ x $ $ u(x, t) $ $ x $ -
Figure 5.
When the initial value is
$ x^2 $ -
Figure 6.
Initial value is
$ x^2 $ $ u(x, t) $ $ t $ -
Figure 7.
Initial value is
$ x^2 $ $ u(x, t) $ $ x $ -
Figure 8.
Initial value is
$ x $ -
Figure 9.
Initial value is
$ x $ $ u(x, t) $ $ t $ -
Figure 10.
Initial value is
$ x $ $ u(x, t) $ $ x $ -
Figure 11.
Initial value is
$ x^2 $ -
Figure 12.
Initial value is
$ x^2 $ $ u(x, t) $ $ t $ -
Figure 13.
Initial value is
$ x^2 $ $ u(x, t) $ $ x $ -
Figure 14.
Loss function with
$ x_{\max}=2 $ -
Figure 15.
$ u(x, t) $ $ x_{\max}=2 $ $ x $ -
Figure 16.
$ u(x, t) $ $ x_{\max}=2 $ $ t $