BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations

Abstract: The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond anything which can be experimentally or analytically validated. In this paper, we show how it is possible to circumvent these limitations by constructing formally-verified neural network solvers for PDEs, with rigorous convergence, stability, and conservation properties, whose correctness can therefore be guaranteed even in extrapolatory regimes. By using the method of characteristics to predict the analytical properties of PDE solutions a priori (even in regions arbitrarily far from the training domain), we show how it is possible to construct rigorous extrapolatory bounds on the worst-case L^inf errors of shallow neural network approximations. Then, by decomposing PDE solutions into compositions of simpler functions, we show how it is possible to compose these shallow neural networks together to form deep architectures, based on ideas from compositional deep learning, in which the large L^inf errors in the approximations have been suppressed. The resulting framework, called BEACONS (Bounded-Error, Algebraically-COmposable Neural Solvers), comprises both an automatic code-generator for the neural solvers themselves, as well as a bespoke automated theorem-proving system for producing machine-checkable certificates of correctness. We apply the framework to a variety of linear and non-linear PDEs, including the linear advection and inviscid Burgers' equations, as well as the full compressible Euler equations, in both 1D and 2D, and illustrate how BEACONS architectures are able to extrapolate solutions far beyond the training data in a reliable and bounded way. Various advantages of the approach over the classical PINN approach are discussed.

• The paper introduces a neural PDE solver framework that delivers formal error bounds, stability assurances, and reliable extrapolation performance.
• It leverages adaptive basis functions and compositional architectures to integrate classical numerical analysis with deep learning techniques.
• Empirical results demonstrate orders-of-magnitude improvements in error metrics and conservation properties over standard neural network approaches.

BEACONS: A Principled Framework for Bounded-Error, Algebraically-Composable Neural PDE Solvers

Motivation and Background

The limitations of conventional neural network (NN) models—mainly their poor out-of-distribution (OOD) generalization—pose significant challenges for scientific machine learning, especially when solving partial differential equations (PDEs) in regimes unattainable by direct simulation or experiment. While classical PDE solvers come with rigorous guarantees on stability, convergence, and conservation, existing neural approaches like Physics-Informed Neural Networks (PINNs) generally lack such formal verification and tend to exhibit degraded extrapolatory performance [raissi_physics-informed_2019, kim_dpm:_2021]. The "BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations" (2602.14853) paper proposes a new neural PDE learning paradigm that delivers formal, machine-checkable guarantees on correctness, stability, and worst-case error—enabling reliable extrapolation and rigorous error bounds even in OOD regimes.

Mathematical Foundations

BEACONS conceptualizes neural networks as a generalization of classical numerical approximators, with flexible, adaptive basis functions. It leverages classical results from nonlinear function approximation theory—specifically, the theorems of Mhaskar and Pinkus [mhaskar_neural_1996, pinkus_approximation_1999]—which supply upper bounds on $L^\infty$ errors for neural approximations of $C^n$-smooth functions:

$$\inf_{u_{\theta} \in \mathcal{V}_{\text{NN}}} \| f - u_{\theta} \|_{\infty} = \mathcal{O}(N^{-n/d})$$

where $N$ is the number of neurons, $n$ is smoothness, and $d$ is the input dimensionality. BEACONS exploits the method of characteristics for hyperbolic PDEs to a priori determine the local solution's smoothness, even in extrapolatory domains, yielding extrapolatory (not merely interpolatory) error bounds. Importantly, unlike PINNs, BEACONS formalizes the conditions under which such error guarantees still hold in the presence of shocks or discontinuities (as for solutions of Burgers’ or Euler equations).

Within BEACONS, solutions to hyperbolic PDEs are composed via shallow networks for local, smooth solution components. For discontinuous solutions, BEACONS constructs compositions (i.e., deep architectures assembled from shallow modules for smooth/discontinuous parts) where error propagation is rigorously bounded. The work proves a key compositionality proposition: if $f$ is $L$-Lipschitz and $g$ is arbitrary, then neural approximations $\tilde{f}$ and $\tilde{g}$ yield

$$\| f \circ g - \tilde{f} \circ \tilde{g} \|_\infty \leq e_f + L e_g$$

This framework generalizes and abstracts the role of flux limiters and TVD schemes in classical numerical analysis.

Automated Formal Verification and Implementation

The BEACONS software stack comprises three components:

• Specification DSL: A Racket-based domain-specific language for encoding hyperbolic PDE systems and their compositional neural solver architectures;
• Automatic Code Generation: Generates optimized C code for producing certified training data, constructing NN architectures, and orchestrating training/validation, interfacing seamlessly with verified classical solvers and multi-physics frameworks (e.g., Gkeyll);
• Automated Theorem Prover: Produces executable, machine-checkable formal proofs certifying solver correctness, including $L^\infty$ error bounds, conservation, and stability, using a symbolic rewriting engine that respects IEEE-754 algebraic guarantees.

The theorem prover decomposes numerical solutions into compositional structures, analyzes smoothness/regularity through characteristics or eigenstructure, and automates the search for optimal NN compositions minimizing error bounds.

Numerical Results and Empirical Validation

BEACONS architectures were benchmarked against conventional fully-connected NNs of equivalent size on a suite of canonical PDE problems:

• Linear advection (1D/2D), inviscid Burgers' (1D/2D), and compressible Euler equations (1D/2D).

In all cases, BEACONS achieved:

• Strictly lower $L^2$ and $L^\infty$ errors (final and integrated), often by orders of magnitude versus standard NNs;
• Significantly improved conservation properties and qualitative feature preservation across long extrapolatory intervals;
• Error realization well within formal worst-case bounds proven a priori, even when extrapolating far beyond the convex hull of training data.

Importantly, BEACONS architectures not only preserved sharp features, shocks, and nonlinear wave interactions, but also accurately tracked advection speeds and rarefaction propagation. In contrast, standard NNs displayed severe diffusion, instability, loss of qualitative structure, and large conservation errors. The formal certificates remained valid as long as the training data, either from verified or high-resolution solvers, were correct with respect to the underlying PDE systems.

Implications and Future Directions

BEACONS establishes a new standard for deploying neural PDE solvers in scientific applications requiring rigorous correctness guarantees. Its formal neurosymbolic paradigm presents strong theoretical and practical advancements over both classical numerical codes and unconstrained neural methods:

• Reliability: Practitioners can obtain formal, machine-checkable guarantees for correctness, stability, and error bounds, closing the gap between scientific ML and computational physics best practices.
• Composability: Algebraic network composition strategies allow deep, expressive models to be built systematically from bounded, analyzable modules, enabling reliable deep architectures.
• Extrapolation: BEACONS is not constrained by training data distributions; solutions in previously unsimulable or computationally intractable regimes become accessible with bounded error.
• Toolchain Integration: Open-source implementation and integration with the formal verification pipelines and classical solver infrastructure enable seamless, certified deployment in large-scale simulation codes.

The work points toward foundational models for multi-physics problems, constructed by assembling and finetuning verified BEACONS "experts" with mixture-of-experts architectures [mu2026comprehensivesurveymixtureofexpertsalgorithms]. The extension to elliptic, parabolic, and ODE problems is straightforward, with the use of appropriate mathematical regularity machinery (e.g., elliptic regularity), and applications in PDE-constrained optimization and inverse problems are immediate next steps.

The BEACONS approach underscores the necessity of rigorous, mathematical underpinnings in neural scientific computing, suggesting that future high-stakes applications of AI in computational science will increasingly demand such formal certification frameworks.

Conclusion

"BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations" (2602.14853) rigorously advances the state of neural PDE solvers by fusing classical numerical analysis and modern deep learning with automated, formal verification. It delivers practically relevant, theoretically justified, and provably reliable neural solvers, establishing a template for future research in certified scientific machine learning and formal neurosymbolic AI for physics.