Dirac-Free Weak Mixed PINN

Integrate this at the loss layer of a mesh-free neural PDE solver, not as a replacement for an ordinary Transformer or MLP block. Use two MLPs: $u_{\phi}(x)$ for the scalar potential and $\boldsymbol\sigma_{\theta}(x)$ for the modified flux. Sample interior points $x_i$ from the domain while excluding a small ball $B_\varepsilon(x_0)$; never evaluate $\mathbf f_2$ at $x_0$. Also sample batches of vector and scalar test functions, either fixed compactly supported RBF/Fourier functions or small adversarial test networks $\tau_\psi$ and $v_\omega$. For each batch, compute $\mathbf f_2(x_i)=(x_i-x_0)/(2\pi\|x_i-x_0\|^2)$, then form the weak constitutive residual $R_c(\tau)=\operatorname{mean}[A^{-1}(\sigma_\theta+f_2)\cdot\tau-u_\phi\,\operatorname{div}\tau]$ plus the boundary term. Form the conservation residual $R_d(v)=\operatorname{mean}[-\sigma_\theta\cdot\nabla v-f_1v]$ plus its boundary term. Minimize $\mathcal L=\max_\psi R_c(\tau_\psi)^2/\|\tau_\psi\|_{p'}^2+\max_\omega R_d(v_\omega)^2/\|v_\omega\|_{W^{1,p'}}^2+\lambda_b\operatorname{MSE}(u_\phi-g,\partial\Omega)$. In code, estimate the dual norms by batch averages and update the test networks for two or three ascent steps per primal update. The paper supplies the singular field and the appropriate $1<p<2$ norm; the engineer must estimate only quadrature integrals, boundary traces, and optionally the empirical $L^p$ flux error. Use analytic polar integration or importance sampling for the annulus near $x_0$ because $\|f_2\|$ behaves like $1/r$. First test on $[-1,1]^2$ with $A=I$, $f_1=0$, known Dirichlet data from a logarithmic Green solution, and source points both aligned with and far from collocation points. Compare against a vanilla PINN using a Gaussian delta of widths $\varepsilon\in\{0.1,0.05,0.01\}$. Success means lower seed-to-seed variance, stable training as $\varepsilon$ decreases, and decreasing weak residual plus $L^p$ flux error for $p=1.5$; a useful target is comparable accuracy with at least 2x fewer collocation points and no source-location-dependent failure.