Replace the standard squared pointwise PDE residual in an elliptic PINN by its discrete $H^{-1}$ norm. The residual is passed through an inverse Dirichlet Laplacian, reducing the dominance of rapidly varying residual modes and acting as a mathematically specified preconditioner for the PINN training gradients.
Partition a large graph into induced subgraphs and perform most parameter updates using only local subgraphs, interleaving them with inexpensive global updates on a randomly subsampled coarse graph. The coarse correction preserves information about cross-partition dependencies while reducing full-graph message passing and communication cost.
Run an adaptive neural ODE solver once to determine accepted step sizes, then train using a regular fixed-length replay of those steps rather than differentiating through adaptive accept/reject logic. The replay can be fused across a batch of trajectories and differentiated with an ordinary reverse sweep, giving the exact discrete gradient of the replayed solver and predictable GPU work.
Replace the raw HJB residual loss of a neural PDE solver with a parametrix-preconditioned fixed-point target. At each local space-time patch, analytically propagate terminal values and source terms through a Gaussian kernel whose covariance uses a frozen diffusion matrix, while asking the network to learn only the variable-coefficient correction. This should reduce the burden on the network to represent stiff high-frequency diffusion dynamics and improve short-horizon convergence.
Attach a calibrated risk monitor to intermediate diffusion states and terminate mutations that are likely to violate hard architecture or performance constraints before full decoding and training. This transfers the paper's separation between proposal generation and authoritative external evaluation into an early-stopping controller for expensive neural architecture trials.
Choose the neural operator's input-history length from the measured correlation time of the unresolved closure signal produced by coarse-graining. This avoids under-memory, which causes systematic closure error, and over-memory, which increases attention cost and can destabilize training. The same diagnostic can drive adaptive memory truncation across physical regimes.