PDEBench-Lang: Representation Effects in Neural Symbolic PDE Reasoning

This project studies whether the representation format of a partial differential equation changes how well a language model can reason about it. Instead of treating PDEs only as numerical objects, the project frames them as structured symbolic language and asks whether formats such as Postfix, LaTeX, Prefix, and natural language lead to different reasoning behavior.

The benchmark, PDEBench-Lang, is built around five canonical PDE families: Heat, Wave, Burgers, Laplace, and Advection. For each generated equation instance, the system converts the same PDE into four symbolic dialects and trains sequence-to-sequence models to predict:

• a structured reasoning chain
• the PDE family label
• a pruned symbolic operator subset for downstream solving

The core research question is whether representation choice affects symbolic pruning quality, family classification accuracy, and reasoning fidelity. To evaluate this, the project introduces a metric called Trash Score, which measures how often a model gives the correct family label while relying on structurally incorrect reasoning.

An example PDE such as

is represented in four different forms:

• Postfix: u t d 0.5 u x x d d * =
• Raw LaTeX: u_{t}=0.5\,u_{xx}
• Prefix: = d(u,t) * (0.5, d(d(u,x),x))
• Natural language: “The time derivative of u equals one-half the second spatial derivative of u.”

The modeling pipeline fine-tunes encoder-decoder language models to map a PDE representation into structured outputs that describe the underlying dynamics. Preliminary experiments with T5-small showed near-perfect scores on the initial natural-language dataset, which led to an important finding: the synthetic benchmark was still too templated and too easy. That result motivated the next research direction of adding richer phrasing variation, greater structural diversity, and more ambiguous cross-family cases.

My contribution to this team project focused on cross-dialect evaluation and benchmarking, helping compare representation formats and analyze how closely model reasoning aligned with the true symbolic structure of the equation.

Main components include:

• synthetic dataset generation across multiple PDE families
• conversion of each PDE into four symbolic dialects
• sequence-to-sequence fine-tuning for reasoning and operator prediction
• benchmarking of pruning quality, label accuracy, and reasoning fidelity
• analysis of representation alignment between symbolic format and LLM pretraining

GitHub repository:
https://github.com/RaghavKrishn/Nlp-group-final-project