Physical-Adjoint Dual-Weighted PINN

Integrate this into a PINN or neural operator for a linear elliptic PDE with a nontrivial target output, such as average temperature over a subregion or boundary flux. Use four network heads, or two networks: primal heads $u_\theta(x),q_\theta(x)$ and adjoint heads $z_\phi(x),r_\phi(x)$. At every collocation point, use automatic differentiation to compute the first-order primal residual $R_p$, physical boundary residual $B_p$, adjoint residual $R_d$, and adjoint boundary residual $B_d$. The adjoint PDE must be derived from the selected output: for a linear operator $Au=f$ and linear QoI $J(u)=\langle j,u\rangle$, solve $A^*z=j$ with boundary conditions obtained by integration by parts; encode this directly as the adjoint residual rather than differentiating through an arbitrary least-squares objective. Train with $L=\eta_p^2+\lambda_d\eta_d^2$ using Adam, alternating primal and adjoint updates or updating both networks jointly. Every $M$ steps, partition the domain into cells, evaluate Monte Carlo estimates of $\eta_{p,K}^2$ and $\eta_{d,K}^2$, compute $\rho_K$ from the paper's balanced formula, retain the top $q$ fraction of cells, and resample collocation points preferentially there while preserving a uniform exploration fraction. Estimate $C$ empirically only for diagnostics; the algorithm does not require it. First test on a 2-D Poisson problem with mixed Dirichlet/Neumann boundaries and a localized volume or boundary-flux QoI, using a 5-layer MLP and 2,000 initial points. Compare uniform PINN residual sampling, primal-only adaptive sampling, and this dual-weighted scheme at equal PDE residual evaluations. Success is lower absolute QoI error $|J(u_\theta)-J(u_{\mathrm{ref}})|$ and faster reduction of that error, ideally by at least 2x at fixed collocation budget, even if the global $L^2$ solution error is unchanged.