README.md

2026GHJ

Codex/ChatGPT (July 2026)

Summary

BEACONS treats neural PDE surrogates as numerical methods that should carry explicit error guarantees. For homogeneous first-order hyperbolic conservation laws, the method of characteristics predicts solution smoothness and shock formation; combined with shallow-network approximation theory, this yields worst-case $L^\infty$ bounds that extend beyond the training interval. To handle discontinuities, the framework decomposes a solution into simpler functions and composes separately trained shallow networks. Its key inequality bounds composition error by $e_f+Le_g$, allowing a smooth outer function with Lipschitz constant $L$ to suppress error from a discontinuous inner approximation. A Racket domain-specific language specifies PDEs and architectures, generates C code for training and inference, and produces executable proof certificates through IEEE-754-aware symbolic rewriting. Deterministic tests train on the first third of simulations and extrapolate through the remainder for one- and two-dimensional advection, inviscid Burgers, and compressible Euler problems. Against equal-sized fully connected networks, BEACONS generally better preserves waves and conserved quantities, has lower $L^2$ and $L^\infty$ errors, and stays within its certified worst-case bounds. The evidence is limited to selected benchmarks; moreover, Euler certificates are conditional because their reference solver is not formally verified, unlike the advection and Burgers solvers.

Contributions

1. Combined characteristic analysis with neural approximation theory to derive extrapolatory worst-case $L^\infty$ bounds for hyperbolic PDE solutions.
2. Established an algebraic composition rule that propagates layerwise approximation errors and motivates bounded-error deep architectures for discontinuous solutions.
3. Implemented a Racket-based specification, C code-generation, training, and validation pipeline for BEACONS solvers.
4. Generated machine-checkable error certificates using a terminating symbolic rewrite system designed to respect IEEE-754 floating-point algebra.
5. Demonstrated improved extrapolation, conservation, and error behavior over equal-sized fully connected networks on one- and two-dimensional advection, Burgers, and Euler benchmarks.