Agentic Retrieval and Reinforcement Learned Equation Chains: A Controlled Generation Framework for Complex and Novel Physics Word Problems

Generating physics diagrams from text requires strict adherence to physical laws. While current generative models produce visually plausible outputs, they systematically hallucinate force vectors, ignore conservation laws, and violate geometric constraints. We present PhyDrawGen, a neuro-symbolic pipeline that decouples semantic scene understanding from physical constraint satisfaction. First, a large language model extracts a typed scene graph from the problem text. A deterministic solver then converts this graph into a Planar Straight-Line Graph (PSLG), encoding force balance, optical paths, and field topologies as exact geometric primitives. Finally, a fine-tuned Qwen-VL model implements a visually grounded propose-verify loop to iteratively correct any constraint violations. Evaluated on a benchmark of 1,449 problems spanning mechanics, optics, and electromagnetism, PhyDrawGen significantly outperforms GPT-5-image, Gemini 2.5 Flash, and Gemini 3 Pro, demonstrating robust physical accuracy even on unusual-object problems.

Physics reasoning fails structurally in small language models: an error at any step propagates forward, corrupting every inference that follows. Limited domain knowledge, hallucination under multi-step derivation, and distributional sensitivity compound this failure. We propose a step-level reward framework that identifies the first reasoning error, generates targeted structured feedback, and trains the model to revise its solution via policy gradient with KL regularization, without exposing it to ground truth solutions as generation targets. Unlike annotation-dependent step-level methods, no preference data construction is required and the external verifier operates exclusively at training time. Across five physics benchmarks, our framework delivers accuracy gains of 17-20% over CoT prompting and 10-16% over the strongest baseline, reduces calculation errors from 56.9% to 23.5%, and reduces miscomprehension errors from 22.3% to 12.0% in the best observed cases. Conceptual errors reduce from 89.7% to 68.7%, yet persist as the hardest failure mode across all conditions.

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.