Second-Order Brownian Jet Residual

1. Integration point: use this as the training loss for a scalar MLP $U_\theta(t,x)$ in a high-dimensional parabolic PDE solver. Do not create separate networks for value, gradient, or Hessian; obtain $Y_i$, $Z_i$, and $\Gamma_i$ with automatic differentiation. The Hessian can be represented as a full matrix for small dimensions, or computed through Hessian-vector products when the dimension is large. 2. Pseudocode: sample $X_0$ from the selected initial distribution. For every trajectory and every grid index $i$, sample $\Delta W_i\sim\mathcal{N}(0,hI_d)$, set $X_{i+1}=X_i+\Delta W_i$, compute $Y_i=U_\theta(t_i,X_i)$, $Z_i=\nabla_xU_\theta(t_i,X_i)$, and $\Gamma_i=D_x^2U_\theta(t_i,X_i)$. Compute $Y_{i+1}=U_\theta(t_{i+1},X_{i+1})$ and $Q_i=\Delta W_i\Delta W_i^\top-hI_d$. Evaluate $f(t_i,X_i,Y_i,Z_i,\Gamma_i)$, form $R_i$ from the displayed formula, average $R_i^2$ over trajectories and time steps, add the terminal value and gradient penalties, backpropagate, and update $\theta$ with AdamW. 3. The mathematics determines the Brownian transition, centered quadratic probe, and residual; expectations are estimated by minibatch averages. Compute $\nabla g$ analytically for manufactured problems or by autodifferentiating the known terminal function. Log separate value, gradient, Hessian, terminal, and residual losses. 4. First experiment: solve a 20- or 50-dimensional manufactured PDE with known smooth solution, comparing this objective against a standard pointwise PINN residual and a first-order stochastic BSDE loss at equal network size and sampled function evaluations. Sweep $N\in\{10,20,40\}$ and measure value, gradient, and Hessian errors on held-out Brownian states. Success means lower full-jet error at equal wall-clock cost, decreasing error as $h$ decreases, and less instability when $f$ has moderate Hessian dependence.