Deep operator networks can become statistically unstable when partial differential equation inputs are observed at thousands of strongly correlated sensors but only a small number of operator samples is available. We introduce FAST-DeepONet, a branch representation combining a fixed spectral path with a regularized projection of the orthogonal residual, in which the directional penalty acts on the effective residual map after each of its rows is normalized. On Navier--Stokes flow a plain DeepONet degrades from $0.0394$ to $0.1556$ mean relative $L_2$ error as the branch grows from $129$ to $8193$ coordinates, while FAST-DeepONet stays near $0.04$, so the sensor grid can be refined without a statistical penalty. Across independent test sets for Navier--Stokes flow, Darcy flow, and signed terminal wavefield prediction it lowers mean relative $L_2$ error by $4.7\%$ to $37.0\%$ with three to seven times fewer trainable parameters. A spectral-only branch sharing the same basis separates the two paths: the fixed spectral path carries the improvement on Navier--Stokes and Darcy, while terminal wave prediction requires the residual path together with its directional penalty. FAST-DeepONet targets coordinate-query architectures and trains on solution values alone.
Evolutionary deep neural networks (EDNNs) solve time-dependent partial differential equations by evolving the neural-network parameters sequentially in time through a local least-squares problem. Their main computational bottleneck is that each time step requires the solution of a dense linear system whose dimension equals the total number of trainable parameters. We propose a low-rank evolutionary deep neural network (LR-EDNN) method that reduces this cost through adaptive tangent-space projection. This construction replaces direct bilinear low-rank factor evolution by a linear reduced problem while preserving the sequential-in-time structure of EDNN. We construct the reduced Jacobian directly through layerwise Jacobian-vector products, without forming the full Jacobian. We further establish a finite-time comparison estimate: the deviation of the LR-EDNN trajectory from full EDNN is bounded by a discrete Gr\"onwall accumulation of the local tangent-space projection defects, with amplification governed by the assumed Lipschitz and directional-coercivity constants. Numerical experiments on a porous-medium equation with drift, one- and two-dimensional Allen-Cahn equations, and two-dimensional viscous Burgers'equations demonstrate that LR-EDNN substantially reduces computational cost while maintaining the accuracy and qualitative fidelity of the full EDNN solver when the rank is chosen adequately.