Large language models (LLMs) bear promise as a fast and accurate material modeling paradigm for evaluation, analysis and design. Their vast number of trainable parameters necessitates a wealth of data to achieve accuracy and mitigate overfitting. However, experimental measurements are often limited and costly to obtain in sufficient quantities for fine-tuning. To this end, here we present a physics-based training pipeline that tackles the pathology of data scarcity. The core enabler is a physics-based modeling framework that generates a multitude of synthetic data to align the LLM to a physically consistent initial state before fine-tuning. Our framework features a two-phase training strategy: utilizing the large-in-amount but less accurate synthetic data for supervised pretraining, and fine-tuning the phase-1 model with limited experimental data. We empirically demonstrate that supervised pretraining is vital to obtaining accurate fine-tuned LLMs, via the lens of learning polymer flammability metrics where cone calorimeter data are sparse.