stat.MLOct 27, 2026

QD-PINN:A Physics-Informed Neural Network Approach to Bayesian Quickest Detection in High-Dimensional Diffusions

Authors: Hang Ye, Yi Liu, Haipeng Xing

Organizations: Department of Applied Mathematics and Statistics, Stony Brook University

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Abstract

Quickest detection problems (QDPs) arise in a wide range of applications. However, the complementarity structure of the associated variational inequalities poses substantial challenges for traditional numerical methods, particularly in high-dimensional settings. We propose a PINN-based framework for estimating the optimal stopping boundary in QDPs with high-dimensional diffusion processes and spatial dependence, while overcoming the curse of dimensionality. Theoretically, we establish a novel equivalence between the free-boundary PDE and an infinite-dimensional stochastic optimization problem, rigorously justifying the use of finite-dimensional neural networks to approximate the true solution. Furthermore, we assess the robustness of the proposed framework by quantifying uncertainty in the learned stopping boundaries using deep ensembles and Monte Carlo dropout. The results show low variability across independent training runs and stochastic forward passes, indicating stable and consistent boundary estimates. We also compare our PINN results with reference numerical solutions where available and find close agreement.

Figures & tables

Figure 1: Training loss (top-left) and S^(x)\widehat{S}(\mathbf{x}) (remaining panels) in Example 5.1 .
Figure 2: Validation total loss (top-left) and S^(x)\widehat{S}({\bf x}) (remaining panels) in Example 5.2 .
Figure 3: Validation total loss (top-left) and S^(x)\widehat{S}({\bf x}) (remaining panels) in Example 5.3 .
Figure 4: Validation total loss (top-left) and S^(x)\widehat{S}({\bf x}) (remaining panels) in Example 5.4 .
Figure 5: Uncertainty quantification results for the Wiener process (top) and the OU process (bottom).
QuantityMean95th percentileMaximum
S^(x)\widehat{S}(\mathbf{x})0.0007100.0007120.000712
V^(π,x)\widehat{V}(\pi,\mathbf{x})0.0098860.0133690.013690
Table 1: Statistics of pointwise standard deviations in Monte Carlo dropout for Example 5.1 ( n=1n=1 ).
Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Dual architecture neural network used to approximate the value function and stopping boundary.See Section 4.2 for details.
Figure 8
SeedMeanMedianQ0.05Q_{0.05}Q0.25Q_{0.25}Q0.75Q_{0.75}Q0.95Q_{0.95}Max
1234560.0068840.0050400.0004400.0022900.0095130.0191500.050168
12340.0058310.0045490.0003770.0021070.0081000.0158490.030691
123456780.0058210.0045040.0003930.0019540.0082610.0158110.041734
20230.0058950.0048190.0004480.0021890.0081920.0151340.050772
20270.0056840.0045600.0004060.0020930.0079160.0149730.041095
123450.0055660.0044040.0004210.0020940.0077700.0146990.034935
Appendix
Table 2: Seed-wise summary statistics of absolute boundary deviations for the OU setting in Example 5.3 . QpQ_{p} denotes the empirical pp -quantile across evaluation states.
nnSIVPS_{\mathrm{IVP}}
10.569022138974
50.562006378463
500.560563307692
1000.560482896107
5000.560418548434
Appendix
Table 3: Numerical reference thresholds for the tridiagonal OU long-term mean change configuration, obtained using the initial-value formulation.
nnSIVPS_{\mathrm{IVP}}SBVPS_{\mathrm{BVP}}∣SIVP−SBVP∣|S_{\mathrm{IVP}}-S_{\mathrm{BVP}}|
10.5690221389740.5690221389733.0098×10−133.0098\times 10^{-13}
50.5620063784630.5620063784632.5302×10−132.5302\times 10^{-13}
500.5605633076920.5605633076922.6534×10−132.6534\times 10^{-13}
1000.5604828961070.5604828961073.0465×10−133.0465\times 10^{-13}
5000.5604185484340.5604185484332.8877×10−132.8877\times 10^{-13}
Appendix
Table 4: Comparison of the IVP and BVP reference thresholds for the tridiagonal configuration. Absolute differences are computed before rounding the displayed thresholds.
nnS‾PINN\overline{S}_{\mathrm{PINN}}EIVPE_{\mathrm{IVP}}EBVPE_{\mathrm{BVP}}Emax⁡E_{\max}
50.5629929.86×10−49.86\times 10^{-4}9.86×10−49.86\times 10^{-4}2.44×10−32.44\times 10^{-3}
500.5614578.94×10−48.94\times 10^{-4}8.94×10−48.94\times 10^{-4}2.42×10−32.42\times 10^{-3}
1000.5614009.17×10−49.17\times 10^{-4}9.17×10−49.17\times 10^{-4}1.94×10−31.94\times 10^{-3}
5000.5612658.46×10−48.46\times 10^{-4}8.46×10−48.46\times 10^{-4}2.15×10−32.15\times 10^{-3}
Appendix
Table 5: Comparison of PINN boundary predictions with the numerical benchmarks
Figure 9: With n=1,000 in Example 5.1
SettingValue
Value network3×1283\times 128 , Tanh
Boundary network2×1282\times 128 , Tanh
OptimizerAdam
Minimum learning rate10−610^{-6}
Iterations ( n≤500n\leq 500 )50,00050{,}000
Iterations ( n>500n>500 )≥80,000\geq 80{,}000
Appendix
Table 6: Main training and implementation parameters used in the numerical experiments.
 πi=Sϕ(xi)+ξi,ξi∼N(0,0.032),\pi_{i}=S_{\phi}(x_{i})+\xi_{i},\qquad\xi_{i}\sim\mathcal{N}(0,0.03^{2}), 
Appendix
Algorithm 1 Residual-based adaptive sampling
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