cs.AIFeb 19, 2026

AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing

Authors: Jianda Du, Youran Sun, Haizhao Yang

Organizations: Department of Mathematics, University of Maryland, College Park, MD, USA · Department of Computer Science, University of Maryland, College Park, MD, USA

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Abstract

PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often demand high computational cost and suffer from limited interpretability. We introduce AutoNumerics, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions. Unlike black-box neural solvers, our framework generates transparent solvers grounded in classical numerical analysis. We introduce a coarse-to-fine execution strategy and a residual-based self-verification mechanism. Experiments on 24 canonical and real-world PDE problems demonstrate that AutoNumerics achieves competitive or superior accuracy compared to existing neural and LLM-based baselines, and correctly selects numerical schemes based on PDE structural properties, suggesting its viability as an accessible paradigm for automated PDE solving.

Figures & tables

Figure 1: The AutoNumerics pipeline. Steps 1–4 handle problem formulation and plan selection. Step 5 implements the coarse-to-fine execution strategy with Fresh Restart logic. Steps 6–7 perform verification and theoretical analysis.
nRMSE ( ↓\downarrow )AdvectionBurgersReact-DiffCNSDarcyGeom. Mean
U-Net5.00×10−25.00\times 10^{-2}2.20×10−12.20\times 10^{-1}6.00×10−36.00\times 10^{-3}3.60×10−13.60\times 10^{-1}−-−-
FNO7.70×10−37.70\times 10^{-3}7.80×10−37.80\times 10^{-3}1.40×10−31.40\times 10^{-3}9.50×10−29.50\times 10^{-2}9.80×10−39.80\times 10^{-3}9.52×10−39.52\times 10^{-3}
PINN7.80×10−37.80\times 10^{-3}8.50×10−18.50\times 10^{-1}8.00×10−28.00\times 10^{-2}−-−-−-
ORCA9.80×10−39.80\times 10^{-3}1.20×10−21.20\times 10^{-2}3.00×10−33.00\times 10^{-3}6.20×10−26.20\times 10^{-2}−-−-
PDEformer4.30×10−34.30\times 10^{-3}1.46×10−21.46\times 10^{-2}−-−-−-−-
UPS2.20×10−32.20\times 10^{-3}3.73×10−23.73\times 10^{-2}5.57×10−25.57\times 10^{-2}4.50×10−34.50\times 10^{-3}−-−-
Table 1: nRMSE (normalized root mean square error) comparison with neural network baselines and CodePDE. All LLM-based methods (CodePDE and Ours) use GPT-4.1. CodePDE results are obtained under the Reasoning + Debugging + Refinement setting (best of 12).
Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

PDEDimErrorRuntime (s)
Explicit analytic solution available (Relative L2L_{2} error)
Advection21.13×10−131.13\times 10^{-13}29.8
Allen-Cahn12.23×10−42.23\times 10^{-4}19.8
Biharmonic26.14×10−16.14\times 10^{-1}89.3
Convection Diffusion28.57×10−38.57\times 10^{-3}34.6
Euler15.21×10−145.21\times 10^{-14}26.0
Appendix
Table 2: Evaluation of proposed framework across 24 benchmark PDEs. The upper block reports relative L2L_{2} error for problems with known analytic solutions; the lower block reports relative residual error.
PlanScoreMethodRationale (summary)
Spectral (RK4, high-res)90Spectral FourierOptimal for smooth periodic advection
FD (WENO3+RK3, high-res)85Finite DifferenceHigh-order upwind, conservative
Spectral (ETDRK4, med-res)80Spectral FourierGood accuracy-cost balance
FV (MUSCL+RK2, med-res)75Finite VolumeLimiter controls oscillations
FD (semi-Lagrangian, med-res)70Finite DifferenceLarge time steps, second-order
FEM (IMEX, med-res)60Finite ElementStable but diffusion treatment irrelevant
Appendix
Table 3: Candidate schemes generated by the Planner Agent and scored by the Selector Agent for the 2D Advection problem.
PlanResidual L2L_{2}Runtime (s)AttemptsRestarts
Spectral (RK4, high-res)1.75×10−31.75\times 10^{-3}23.820
FD (WENO3+RK3, high-res)3.18×1043.18\times 10^{4}57.540
Spectral (ETDRK4, med-res)8.02×10−158.02\times 10^{-15}35.340
FV (MUSCL+RK2, med-res)1.94×10−11.94\times 10^{-1}33.220
FD (semi-Lagrangian, med-res)2.27×10−22.27\times 10^{-2}15.820
Appendix
Table 4: Execution results for the top-5 candidate schemes on the 2D Advection problem.
PDEDim.BCPDE TypeSelected Scheme
Periodic boundary conditions
Advection2PeriodicHyperbolicSpectral Fourier (RK4)
Convection Diffusion2PeriodicParabolicSpectral Fourier (IMEX)
Schrödinger1PeriodicDispersiveSpectral Fourier (Split-Step)
Navier–Stokes2PeriodicParabolicFEM (IMEX)
Shallow Water1PeriodicHyperbolicFD (explicit)
Appendix
Table 5: Numerical schemes selected by the Planner Agent for each benchmark PDE.
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