Lu Zhang

Kui Ren, Lu Zhang

The task of simultaneously reconstructing multiple physical coefficients in partial differential equations (PDEs) from observed data is ubiquitous in applications. In this work, we propose an integrated data-driven and model-based iterative reconstruction framework for such joint inversion problems where additional data on the unknown coefficients are supplemented for better reconstructions. Our method couples the supplementary data with the PDE model to make the data-driven modeling process consistent with the model-based reconstruction procedure. We characterize the impact of learning uncertainty on the joint inversion results for two typical inverse problems. Numerical evidence is provided to demonstrate the feasibility of using data-driven models to improve the joint inversion of multiple coefficients in PDEs.

Kuo Zhou, Lu Zhang

Large Language Models (LLMs) have demonstrated formidable capabilities in solving mathematical problems, yet they may still commit logical reasoning and computational errors during the problem-solving process. Thus, this paper proposes a framework, MATH-VF, which includes a Formalizer and a Critic, for formally verifying the correctness of the solutions generated by large language models. Our framework first utilizes a Formalizer which employs an LLM to translate a natural language solution into a formal context. Afterward, our Critic (which integrates various external tools such as a Computer Algebra System and an SMT solver) evaluates the correctness of each statement within the formal context, and when a statement is incorrect, our Critic provides corrective feedback. We empirically investigate the effectiveness of MATH-VF in two scenarios: 1) Verification: MATH-VF is utilized to determine the correctness of a solution to a given problem. 2) Refinement: When MATH-VF identifies errors in the solution generated by an LLM-based solution generator for a given problem, it submits the corrective suggestions proposed by the Critic to the solution generator to regenerate the solution. We evaluate our framework on widely used mathematical benchmarks: MATH500 and ProcessBench, demonstrating the superiority of our approach over existing approaches.