# Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards

arXiv: https://arxiv.org/abs/2607.10474 · HTML: https://mlsanity.com/paper/2607.10474/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 78 references that are in the map.
- First posted on arXiv 11 Jul 2026.

## Builds on (most cited first, 30 of 38)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [Measuring Coding Challenge Competence With APPS](https://mlsanity.com/paper/2105.09938/) (2021)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [CodeRL: Mastering Code Generation through Pretrained Models and Deep Reinforcement Learning](https://mlsanity.com/paper/2207.01780/) (2022)
- [SciCode: A Research Coding Benchmark Curated by Scientists](https://mlsanity.com/paper/2407.13168/) (2024)
- [RLEF: Grounding Code LLMs in Execution Feedback with Reinforcement Learning](https://mlsanity.com/paper/2410.02089/) (2024)
- [Beyond Binary Rewards: Training LMs to Reason About Their Uncertainty](https://mlsanity.com/paper/2507.16806/) (2025)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [MetaOpenFOAM: an LLM-based multi-agent framework for CFD](https://mlsanity.com/paper/2407.21320/) (2024)
- [CodePDE: An Inference Framework for LLM-driven PDE Solver Generation](https://mlsanity.com/paper/2505.08783/) (2025)
- [PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs](https://mlsanity.com/paper/2502.00963/) (2025)
- [FEABench: Evaluating Language Models on Multiphysics Reasoning Ability](https://mlsanity.com/paper/2504.06260/) (2025)
- [PINNsAgent: Automated PDE Surrogation with Large Language Models](https://mlsanity.com/paper/2501.12053/) (2025)
- [MooseAgent: A LLM Based Multi-agent Framework for Automating Moose Simulation](https://mlsanity.com/paper/2504.08621/) (2025)
- [ALL-FEM: Agentic Large Language models Fine-tuned for Finite Element Methods](https://mlsanity.com/paper/2603.21011/) (2026)

## Built on by (most cited first, 2 of 2)

- [NeutronGym: Physics-Graded Neutron Instrument Design for LLM Agents](https://mlsanity.com/paper/2610.03631/) (2026)
- [Training Numerical Intelligence via Auto-Diagnosis and Skill Discovery](https://mlsanity.com/paper/2610.03872/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert knowledge of discretization schemes, stability conditions, and boundary treatments. Recent work has begun to frame PDE solving as a code-generation task for large language models (LLMs), yet existing approaches operate primarily at inference time: relying on prompting, debugging, self-refinement, and test-time scaling rather than adapting the model itself. In parallel, reinforcement learning with verifiable rewards has emerged as a post-training paradigm for code and math reasoning, but its verifiers are typically binary: a compiler runs, or a test passes. Such signals discard the graded structure of scientific correctness, where two solvers may both execute and yet differ in solution accuracy by orders of magnitude. In this work, we introduce RLVP: Reinforcement Learning with Verifiable Physics, an RL post-training framework for multi-PDE solver code generation. RLVP addresses this verifiability gap with a hybrid verifier: hard program-validity checks ensure executability, while continuous physics rewards score function-space accuracy and PDE-residual consistency. A single policy is post-trained across diverse PDE families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems. RLVP improves over both pre-trained and supervised-only baselines on PDE benchmarks, and shows zero-shot improvement transfer to held-out PDEs. We show that a smaller LLM post-trained with RLVP can outperform prompting a frontier model on in-distribution PDE solver generation. The trained policy shows evidence of compositionality in numerical motifs: it recombines stencils, time-stepping schemes, and boundary-handling primitives learned from the PDEs used in training into generated solvers for unseen PDE problems.