Finite-Difference Conditional Neural Operator

(1) Integration point: implement a conditional value-function module for an RL/world-model task or a solver for a known dynamic-programming PDE. Let x2 be a fixed-length vector such as a histogram, empirical-measure embedding, population statistic, or set-encoder output. Let x1 be q1<=4 continuous state coordinates with a rectangular grid of J points. Use an MLP or DeepSets encoder E_phi(x2), concatenate its embedding with a learned positional encoding of every x1 grid point, and decode all grid values v=G_theta(x2) in one forward pass. (2) Pseudocode: sample a batch of x2 and PDE parameters; compute v=G_theta(x2); reshape v to the x1 grid; compute central differences D_i v and one-sided boundary differences; compute x2 derivatives by autograd through x2, or estimate a coordinate r by [G_theta(x2+epsilon e_r)-G_theta(x2-epsilon e_r)]/(2 epsilon); evaluate R_j=F(x1_j,x2,v_j,Dv_j,dVdx2); add boundary loss and the signed shape penalty; backpropagate L and apply AdamW. (3) Compute from the paper's mathematics the finite-difference residual and explicit boundary and shape terms. Estimate grid spacings h_i, derivative step epsilon, PDE coefficients, and shape signs s_i from the task; do not learn them unless no prior is available. Use masked stencils near boundaries and normalize each residual term by its batch standard deviation. (4) First experiment: solve a two-dimensional HJB or Black-Scholes-style PDE whose coefficients depend on a 128-bin distribution vector, comparing a tensor-product MLP, a standard PINN, and this operator with matched parameter counts. Measure PDE residual, boundary error, held-out distribution generalization, and wall-clock time. The expected signal is lower residual and boundary violation at equal compute, with memory scaling approximately linearly in the low-state grid size rather than exponentially in q1+q2.