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

• High-dimensional, high-frequency streams with spatial and cross-channel dependence make real-time multivariate detection costly and traditional methods impractical.Source: §1 · p. 1
• For Bayesian diffusion models, quickest detection is an optimal-stopping problem whose PDE free boundary separates continuation and stopping regions under value-matching, smooth-fit, and complementarity constraints.Source: §1 · pp. 1–2
• The proposed PINN jointly estimates the value function and stopping boundary, enforcing PDE, boundary, value-matching, smooth-fit, and complementarity constraints through a unified loss and specialized high-dimensional sampling.The Kolmogorov transformation derives smooth fit in the original state variables without an additional reduced representation.Source: §1 · p. 2
• Experiments estimate boundaries for Brownian and Ornstein–Uhlenbeck processes up to 1,000 dimensions and examine spatial correlations and uncertainty in the PINN solutions.Source: §1 · p. 2
• An infinite-dimensional stochastic optimization reformulation establishes theoretical equivalence and justifies using finite-dimensional neural networks to approximate high-dimensional quickest-detection solutions.Source: §1 · p. 2

• Simulation-based methods learn stopping decisions or continuation values from trajectories; Becker et al. (2019) use Monte Carlo samples with time discretization, while Herrera et al. (2024) train only the output layer.Source: §2 · pp. 2–3
• PDE-based PINNs solve free-boundary problems without sample paths; examples include Stefan problems and multi-asset American options with jointly approximated value functions and boundaries.Wang and Perdikaris (2021) and Gatta et al. (2023) are cited as examples.Source: §2 · p. 3
• For quickest detection, the framework adds smooth-fit enforcement in the original state space, strict complementarity constraints, uncertainty quantification, and high-dimensional sampling.Source: §2 · p. 3

3.1 Bayesian formulation

• The change time θ has prior mass π at zero and exponential survival otherwise; correlated Wiener observations follow a diffusion with a drift change, under stated coefficient and covariance assumptions.Source: §3.1 · p. 3

3.2 Optimal stopping and free-boundary

• In the general setup, stopping minimizes expected pre-stop observation costs plus false-alarm loss; the classical Bayesian risk combines false-alarm probability with expected delay cost.Source: §3.2 · p. 4
• The value function satisfies a variational inequality, with continuation region π < S(x) and an optimal rule that stops when the posterior reaches S(x).The value function is characterized as the unique viscosity solution.Source: §3.2 · pp. 4–5
• The Kolmogorov transformation supplies a smooth-fit condition to obtain a unique solution, and the resulting stopping boundary is required to be continuous and of bounded variation.Source: §3.2 · p. 5

4 Physical Informed Neural Network

• The unified loss embeds PDE, boundary, value-matching, smooth-fit, and complementarity constraints; residuals address degenerate solutions that assign nearly the whole domain to one region.The Fischer-Burmeister function is zero exactly when its arguments are nonnegative and have product zero.Source: §4 · p. 5, §4.1 · p. 6

4.1 Infinite dimensional stochastic optimization

• Under smoothness, positive-density, and square-integrability assumptions, the stochastic optimization characterization is equivalent to a classical solution of the free-boundary problem.The theorem states the equivalence for n ≥ 1.Source: §4.1 · p. 6
• The value function and stopping boundary form the unique global minimizer, and replacing function spaces with sufficiently large neural-network classes yields a finite-dimensional optimization problem.The formulation jointly targets V(π, x) and S(x).Source: §4.1 · p. 6

4.2 Neural network approximation

• Two separate feedforward networks approximate the value function V(π, x) and the unknown stopping boundary S(x).The networks have separate parameter sets.Source: §4.2 · p. 6
• Adaptive training combines refined residual-guided points with uniform Sobol samples, while an added continuation-region PDE residual supplies a gradient signal to boundary predictions.The sampling strategy increases resolution near the evolving free boundary while retaining global exploration.Source: §4.2 · p. 7

4.3 Uncertainty Quantification

• Parameter uncertainty reflects finite collocation points, initialization, optimization, architecture, and sampling; deep ensembles use independent runs to form empirical confidence bands.The bands use empirical quantiles across the ensemble approximations.Source: §4.3 · p. 7
• Monte Carlo dropout estimates neural-network variability by holding learned parameters fixed and sampling dropout masks for stochastic predictions.The resulting variability is across dropout realizations rather than independent training runs.Source: §4.3 · p. 7

5.1 Brownian motion with a change in drift

• PINN boundary estimates of 0.557–0.558 compare with the exact Brownian-motion boundary S(x) = 0.5560, with absolute errors on the order of 10^-3.Figure 1 shows training loss and estimated boundaries; the displayed surface varies with the first two coordinates while remaining coordinates are fixed.Source: §5.1 · pp. 8–9, Figure 1 · p. 8

5.2 OU process with a change-point in mean reversion speed

• For OU changes in mean-reversion speed, the n = 1 estimates are U-shaped; at n = 5, the Toeplitz case varies more than the tridiagonal case.For n ≥100, the plotted (x1, x2) slices are partial views, so apparent flatness does not establish state independence; without analytical benchmarks, assessment focuses on topology.Source: §5.2 · p. 10, Figure 2 · p. 10, Figure 3 · p. 11

5.3 OU process with a change-point in long-term mean

• For an OU change in long-term mean, the problem implies a spatially constant boundary for each fixed n, while higher-dimensional plotted slices show minor spatial variations.The n = 1 estimate is nearly 0.5700; higher-dimensional slices are around 0.563 for n = 5 and 0.561 for n = 50, 100, and 500, versus numerical references from 0.5604 to 0.5690.Source: §5.3 · p. 10, Figure 3 · p. 11

5.4 Uncertainty Quantification and Error Analysis

• Across 10 independent runs, the Wiener-process ensemble mean is approximately 0.5578 and its 5th–95th percentile band is narrower than 10^-3; the OU band is on the order of 10^-2.These comparisons use 4096 test points and show variability along displayed x1 slices; the OU results indicate modest, state-dependent variability across runs.Source: §5.4 · p. 11
• For the n = 1 Brownian example, Monte Carlo dropout yields a maximum pointwise boundary standard deviation of 7.12 × 10^-4.The analysis uses 100 stochastic forward passes at each of 4096 fixed test points with dropout probability 0.05.Source: §5.4 · p. 11, Table 1 · p. 11

6 CONCLUSION

• The framework reformulates the high-dimensional free-boundary problem as infinite-dimensional stochastic optimization and jointly approximates the value function and stopping boundary with neural networks.Its loss incorporates the governing PDE, boundary-related conditions, complementarity constraints, and high-dimensional sampling strategies.Source: §6 · p. 12
• Brownian-motion and OU experiments demonstrate accurate approximation of benchmark stopping boundaries and applicability in dimensions up to 1,000.The experiments also accommodate different changes in diffusion dynamics and spatial correlation structures.Source: §6 · p. 12
• Future work includes adaptive sampling and boundary-focused training, extensions to other high-dimensional sequential-analysis problems, and systematic uncertainty and approximation-error analysis.The latter analyses could provide further information on the reliability of learned value functions and stopping boundaries.Source: §6 · p. 12

A.1 Proof of Theorem 4.1

• A classical free-boundary solution makes every nonnegative loss term vanish, so its loss attains the minimum value zero.The argument uses the PDE and variational inequalities in the continuation and stopping regions, plus value matching and smooth fit at the boundary.Source: §A.1 · p. 14
• If the loss is zero, its nonnegative residual terms recover the PDE in the continuation region and the stopping equation in the stopping region.The proof then uses residual inequalities and the Fischer–Burmeister condition to recover the variational inequalities and complementarity.Source: §A.1 · p. 14
• Continuity and regularity extend the boundary residuals from almost everywhere to everywhere, yielding value matching and smooth fit and completing the equivalence.The proof concludes that the free-boundary problem is solved when the loss functional is zero.Source: §A.1 · p. 14

A.2 Detailed Mathematical Formulations of the Numerical Examples 5.1–5.4

• Complete mathematical formulations are provided for the numerical examples considered in Section 5, and the passages specify infinitesimal operators and free-boundary problems for sequential change-point detection.Source: §A.2 · p. 14, §A.2.1 · p. 15, §A.2.2 · p. 15

A.2.1 Example 5.1

• For Example 5.1, the free-boundary equation uses the operator Lbm and is paired with a smooth-fit condition.The equation is stated for π in (0,S(x)).Source: §A.2.1 · p. 15

A.2.2 Example 5.2 and Example 5.3

• For the sequential change-point detection problem, the value function V(π, x) and unknown stopping boundary S(x) are obtained from a free-boundary problem whose continuation equation is Lou,1V(π, x) − rV(π, x) + c0π = 0 for π ∈ (0, S(x)).Source: §A.2.2 · p. 15

B.1 Figure of Dual Architecture Neural Network

• Figure 6 depicts the dual neural-network architecture used to approximate the value function and stopping boundary.Section 4.2 contains further details.Source: Figure 6 · p. 17
• Figure 7 shows uncertainty-quantification results for the Wiener process in Example 5.1.Source: Figure 7 · p. 18
• Figure 8 shows uncertainty-quantification results for the OU process in Example 5.3.Source: Figure 8 · p. 18

B.2 Figure of More Uncertainty Quantification Results

• For the Wiener process, the median pointwise ensemble standard deviation is approximately 2.7×10^-4, with the central 50% between approximately 2.4×10^-4 and 2.9×10^-4.The largest upper-tail values reach approximately 4.1×10^-4; the passage describes the boundary predictions as consistent across independent runs.Source: §B.2 · p. 18
• For Example 5.3, the median pointwise ensemble standard deviation is approximately 5.6×10^-3, with the central 50% from 4.2×10^-3 to 7.4×10^-3.The maximum standard deviation is approximately 2.1×10^-2.Source: §B.2 · p. 18
• Across OU-setting seeds, mean absolute boundary deviations range from approximately 4.4×10^-3 to 6.9×10^-3, and every 95th percentile is below 2.0×10^-2.The deviations compare each model with the mean prediction of the remaining ensemble members and are summarized across evaluation states.Source: Table 2 · p. 18, §B.2 · p. 19

B.3 Numerical benchmark with Example 5.4

• Whitening and rotating the tridiagonal observations isolates a one-dimensional coordinate that retains the change-point information, yielding a boundary constant in x for each fixed n.The scalar setting admits a one-dimensional optimal-stopping formulation; the boundary value depends on n and Rn through qn.Source: §B.3 · pp. 19–20
• The reference calculation retains discount rate r=0.05 and obtains thresholds with independent IVP and BVP formulations.The IVP uses local expansions and a Radau IIA method; the BVP uses fourth-order collocation, and neither initialization uses PINN predictions.Source: §B.3 · p. 20
• The IVP and BVP reference thresholds differ by at most 3.05×10^-13 across five dimensions, providing a numerical consistency check.The two formulations share the reduced model and endpoint regularity condition.Source: Table 4 · p. 21, §B.3 · p. 21
• PINN mean-boundary deviations from the numerical benchmarks are 8.46×10^-4 to 9.86×10^-4 across four tested dimensions, and the largest sampled pointwise error is 2.44×10^-3.The mean predictions are slightly above the reference thresholds; both reported errors remain below 1.0×10^-3 and 2.5×10^-3, respectively.Source: Table 5 · p. 21, §B.3 · p. 21

B.4 With Dimension = 1,000 in Example 5.1

• At n=1,000 in Example 5.1, the learned stopping boundary is approximately 0.557, close to the theoretical result of 0.556.Source: Figure 9 · p. 22, §B.4 · p. 22

C.1 Computational resources and runtime

• Experiments ran on a single workstation with one GPU per training run; dimensions up to 500 used 50,000 iterations, while n=1,000 used 80,000.A typical run took approximately 30–50 minutes, depending on dimension and model configuration.Source: §C.1 · p. 22

C.2 Training hyperparameters

• Separate fully connected networks approximate the value function from posterior probability and state, and the stopping boundary from state alone.The value network has three 128-unit tanh layers; the boundary network has two 128-unit tanh layers.Source: §C.2 · p. 22
• The batch size of 8,192 reflects available computational resources, balancing collocation coverage against GPU memory requirements rather than an optimized choice.Larger collocation sets can provide broader coverage in high-dimensional problems.Source: §C.2 · p. 23
• Hutchinson trace estimation with four probe vectors avoids constructing the full Hessian for second-order PDE residual terms.The stated purpose is to reduce computational and memory costs for high-dimensional operators.Source: §C.2 · p. 23

C.3 Residual-based sampling

• 40,000 spatial candidates are partitioned into bins, where composite-residual sampling favors larger residuals while bin-wise quotas limit far-field domination.The composite score combines loss residuals with boundary-related residuals; bin allocation uses prescribed spatial-bin weights.Source: §C.3 · pp. 23–24
• The 8,192-point batch combines 25% residual-adaptive, 25% local, 40% radius-stratified shell, and 10% full-domain Sobol samples.Posterior samples are drawn near the current boundary with probability 0.5, while remaining samples provide broader posterior coverage.Source: §C.3 · p. 24