Direction-Adaptive Plane-Wave Discontinuous Galerkin Methods for the Helmholtz Equation
arXiv:2608.18380
2026
Architecture
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops a residual-driven adaptive basis in which nonlinear propagation directions are optimized while linear coefficients are eliminated exactly by variable projection. Its transferable asset is the separation of nonlinear dictionary geometry from linear feature fitting, together with Gramian-based removal of numerically dependent features and residual-based local error signals. A practical neural-network analogue is an adaptive complex Fourier-feature layer whose directions are updated by an outer optimizer while its coefficients or readout are solved by weighted least squares. This is most promising for PINNs, neural operators, and coordinate networks representing fields with sparse directional or oscillatory structure.
Ideas from this paper
✗ Mechanism failed
2026
Replace fixed random Fourier features in a coordinate network or PINN with a small complex Fourier dictionary whose propagation directions are learned from a weighted residual. Given directions, solve the linear feature coefficients exactly or by ridge regression, and optimize only the directions in the outer loop. This should represent low-directional-complexity fields with fewer features and avoid wasting gradient updates on coefficients that can be fitted analytically.
Useful7/10
Difficulty5/10
Novelty6/10
Unverified
2026
Use the weighted feature Gramian to remove nearly dependent Fourier, sinusoidal, or dictionary features during training instead of relying only on magnitude pruning. The criterion detects redundant directions even when their learned coefficients are individually large, improving conditioning of the linear readout and reducing inference cost.
Useful6/10
Difficulty4/10
Novelty5/10