Learned PDE solvers often entangle governing operators with the geometry, boundary conditions, and discretization used for training. This limits reuse when the same physics is posed on new domains, and it also makes physical-law discovery geometry-dependent. We introduce Geometry-aware LegONet (gLegONet), a boundary-manifold extension of Lego-like operator learning. Physical mechanisms are pretrained once as modular variational blocks on an ambient spectral domain. For a target geometry, sampled boundary constraints define an affine admissible manifold. Its mass-orthonormal tangent coordinates are used to evolve the dynamics and evaluate candidate law-discovery features directly. Changing the domain therefore changes only an algebraic coordinate interface, not the learned operator blocks. This converts arbitrary-domain PDE learning from geometry-specific retraining or soft penalty enforcement into boundary-guaranteed assembly of reusable mechanisms. In forward simulations and sparse identification tests on unseen domains, the method maintains boundary residuals near the algebraic tolerance and yields predictive governing laws from short-time observations.
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.
