Cell-Averaged Residual Corrector

Use this at training time for a neural ODE, latent dynamics model, or PINN predicting a periodic incompressible velocity u_theta(x,t). Divide every training trajectory into cells of tau=4-16 solver steps. Evaluate the PDE residual r_theta(x,t)=partial_t u_theta+div(u_theta tensor u_theta)-nu Delta u_theta+grad p_theta at quadrature times inside each cell. Compute the cell average with trapezoidal or Simpson quadrature, bar_r_J=(1/tau) sum_m w_m r_m, then compute the cumulative integral q_m=-P sum_{ell<m} Delta t_ell(r_ell-bar_r_J). Here P is implemented by FFT: for nonzero wavevector k, P_hat(k)=I-k k^T/|k|^2, and P_hat(0)=0. Either train the corrected trajectory u_corr=u_theta+alpha q, with alpha initialized to 1, or add the corrected residual r_corr=r_theta+partial_t q; the latter removes the zero-mean component in discrete arithmetic. In pseudocode: r=residual(u,t); rbar=weighted_mean(r); q[0]=0; for m in 1..M: q[m]=q[m-1]-dt[m-1]*P(r[m-1]-rbar); loss=mean(norm(r+dt_q)) plus endpoint and data losses. The zero endpoint property is mathematical; quadrature error, projection error, and the useful alpha are measured empirically. Start with a 2D Taylor-Green-vortex PINN or neural ODE at 64^2, compare against the same network trained with pointwise residual loss and against simple temporal averaging. Success is a lower maximum-in-time residual, reduced energy drift, and equal or better endpoint accuracy at the same number of network evaluations.