Killed-Resolvent Residual for Neural Obstacle Solvers

(1) Integrate into a neural optimal-stopping, American-option, intervention-control, or RL value solver. The network outputs V_theta(x); define the learned continuation set C_theta={x: V_theta(x)>G(x)+margin}. The key operation is not a global PDE residual, but a trajectory residual accumulated only until the simulated state exits C_theta. (2) Pseudocode: sample initial states x0; simulate reflected Euler steps x_{t+1}=Proj_{R_+^2}(x_t+mu(x_t)dt+sigma(x_t)sqrt(dt)xi_t); while V_theta(x_t)>G(x_t)+m and t<T, accumulate q += exp(-r*t*dt)*Gamma_smooth(x_t)*dt; detect u_t=x_{t1}-alpha*x_{t2}; add kink contribution gamma_delta=-(n^T a(x_t)n)/(2*sqrt(1+alpha^2))*delta_eps(u_t), with delta_eps(u)=exp(-u^2/(2eps^2))/(sqrt(2*pi)*eps); stop when the continuation test fails; fit V_theta(x0) to G(x0)-q, or minimize the squared identity residual V_theta(x0)-G(x0)+q. (3) Compute Gamma_smooth=c+rG_smooth-LG_smooth using a smooth surrogate G_smooth=sqrt((x1-alpha*x2)^2+eps^2)+alpha*x2; compute the singular correction explicitly from the displayed formula. Estimate a and reflection by the simulator, while r, c, alpha, dt, and eps are known. Use a target network or lagged C_theta every 1000 updates to avoid moving-target instability. (4) First experiment: solve a two-dimensional reflected Brownian stopping problem with constant covariance and max payoff, using a four-layer MLP and compare global PINN residual training, Monte Carlo stopping-value regression, and the killed-resolvent residual. Use 10,000 trajectories and a fixed validation grid. Success is lower value error near the diagonal and stopping boundary, reduced variance across random seeds, or the same error with at least 30% fewer simulated time steps because trajectories terminate at the learned boundary.