GAV system improves testing and feedback for PDE-solving agents
GAV: A scientific experimentation environment for PDE agents
Machine LearningArtificial Intelligence
Summary
Programs that solve complex equations modeling physical phenomena need good feedback to get better. The authors created GAV, a system that generates new test problems and checks program answers without costly extra calculations. By working backward from known solutions, it creates matching problems to test programs more efficiently. This helps program designers find mistakes and improve their solutions faster. GAV showed big improvements on tasks involving fluid dynamics and plasma physics equations.
What this means in practice
- •For software developers for scientific computing: Integrate GAV to efficiently verify and improve PDE-solving programs without expensive reference computations.
- •For computational fluid dynamics engineers: Use GAV to generate new fluid dynamics test cases and get reliable feedback on simulation program accuracy and cost.
Authors
Muheng Li, Wenlong Mou
Abstract
Reliable feedback is essential for coding agents to develop programs for partial differential equations (PDEs). Yet evaluating these programs on new PDE problems often requires expensive reference computations, while fixed datasets restrict the range of experiments. We introduce Generator-Agent-Verifier (GAV), a scientific experimentation environment that makes new PDE problems verifiable by construction. The Generator learns a distribution of continuous reference solutions using data from numerical simulations or experimental measurements to represent a target physical regime. Using the method of manufactured solutions, it works backward from the generated reference solutions to construct matching PDE problems without a separate numerical solve. The Agent designs experiments, investigates failures, and iteratively improves its program. The Verifier evaluates candidate programs against the reference solutions and provides numerical feedback on accuracy and computational cost. This feedback can also guide the generation of new problems that probe the current program's weaknesses. We provide a theoretical analysis of evaluation reliability and guided development. Experiments on Navier-Stokes and Vlasov-Poisson-type equations across different data sources and regimes demonstrate the effectiveness of GAV. In particular, on the numerical Navier-Stokes task, GAV reduces the mean test error of three existing PDE agents by more than an order of magnitude relative to their original setups. Code is available at https://github.com/PotatoJnny/PDE-GAV.