Source-linked AI summary
What Should a Large Language Model See? Physical Invariants as a Data Representation for PDE Discovery
Fan Yang, Matt Thomson
TL;DR
Theory construction for molecular systems is too slow for the volume of modern field data, and raw spatiotemporal fields are difficult to place directly in LLM prompts. The paper introduces data interpretation, which supplies compact physical measurements to an LLM for PDE symbolic regression. On simulated-field benchmarks, interpreted inputs substantially outperform raw field slices, without training and at low computational cost.
Problem
Conventional theory building is slow and expertise-dependent, while raw spatiotemporal field data generally cannot be placed directly in an LLM prompt.
Method
Data interpretation computes compact, interpretable physical measurements from fields and supplies them directly to an LLM proposing PDEs without a predefined term library.
Results
F1 = 0.720 for data interpretation versus 0.225 for raw field slices, with exact recovery on 14 of 44 samples versus 2.
Takeaways & Limitations
Data interpretation is a compact and useful input for LLM-driven symbolic regression and offers a route toward automated top-down field theory construction.
Takeaways & Limitations
The study uses 44 samples, one frozen model, noise-free simulated fields, and characterization tools that do not determine nonlinear-term forms and are fragile to noise.
Abstract
from arXiv · showhide
Understanding how molecular interactions govern macroscopic behaviour is a central challenge in molecular sciences. However, conventional theory building cannot keep pace with the vast datasets modern experimentation routinely produces. Large language models offer a promising route to automating theory construction, but a spatiotemporal field cannot be directly placed in a prompt. Existing models generally learn about the data only through a score measuring how well each proposal fits it. Here we introduce data interpretation, a stage that measures the field into the quantities a theorist would consult and supplies them to the model as a direct input. On a benchmark of simulated fields, interpretation nearly triples the accuracy of recovered equations relative to showing the raw data, at negligible computational cost and without any training. By allowing a language model to read field data as a theorist does, data interpretation offers a practical route to automated field theory construction that can coevolve with experimentation.
1 Introduction
Automating PDE-based field theory discovery could address the expertise and throughput limits of conventional modeling. The paper proposes representing field data through theorist-relevant interpretations so an LLM can use them directly during symbolic regression.
- Motivation: Months-to-years theory development by specialists cannot match experiments generating thousands of molecular systems.Conventional bottom-up and top-down modeling approaches are both slow and expertise-dependent.
- Background: Sparse-regression methods recover PDEs from predefined candidate libraries, limiting discovery to terms written down in advance.Symbolic regression removes this library constraint but requires combinatorial search.
- Background: LLMs can serve as proposal mechanisms for symbolic regression, while evaluators score candidate equations to guide iterative refinement.Prior systems combine LLM proposals, numerical evaluation, and evolutionary or reflective feedback.
- Motivation: Human theorists infer mechanisms from salient field patterns, such as shock waves indicating convection or Turing patterns signaling diffusion-reaction systems.These inferences rely on interpreted patterns rather than numerical field values alone.
- Contribution: Data interpretation supplies compact, auditable physical measurements directly to the LLM instead of relying only on evaluator scores.The diagnostics replace prohibitively expensive raw-field prompting, require no training, and take a small fraction of one model call.
2 Methodology
The method converts simulated fields into spectral and correlation-based diagnostics, prompts a frozen LLM to propose unrestricted PDEs, then canonicalizes and evaluates them. Measurements probe temporal order, linear coefficients, nonlinearity, and advection while exposing important identifiability limits.
- Pipeline: The PDE-discovery loop runs data interpretation, LLM equation proposal, parsing and canonicalisation, and evaluation with a frozen QwQ-32B model.The model remains unchanged throughout all experiments.
- Data interpretation: Four interpretation questions probe temporal order, linear coefficients, linearity, and advection using measurements derived from field dynamics.Spectral analysis decomposes periodic scalar fields into spatial modes whose temporal behavior informs PDE structure and coefficients.
- Data interpretation: Monotonic versus oscillatory Fourier-mode amplitudes indicate temporal order, but overdamped second-order equations can appear indistinguishable from first-order equations.The interpretation constrains plausible structures rather than uniquely determining one.
- Data interpretation: A regression of modal decay rates can recover linear coefficients, while oscillation frequencies provide an estimate of c2 without using the intercept to determine σ.For first-order dynamics, r(k) relates to ν|k|2−σ; oscillatory modes instead use frequency scaling.
- Data interpretation: Nonlinearity is probed through high-wavenumber spectral-energy transfer and the R2 of a regression assuming independent modal decay.Mode coupling drives energy toward high wavenumbers and lowers the independent-mode regression fit.
- Data interpretation: Translation-based correlation gain and its maximizing shift distinguish directional advection from isotropic diffusion or growth.A nonzero shift estimates advection velocity over the time interval, whereas isotropic evolution leaves the maximum near zero.
- LLM and evaluation: Diagnostics are rendered in about 300 tokens, while the LLM may combine the field and derivatives freely without a predefined candidate-term library.Proposals are extracted into canonical form, then coefficients are fitted by regularized least squares and scored using residual and sparsity penalties.
3 Experiments
The experiments evaluate PDE discovery on simulated scalar fields using controlled input representations, metrics, and benchmark strata. Data interpretation substantially outperforms raw field slices and performs best when its characterization is most complete.
- 3.1 Data and benchmark: Fields are generated by numerically solving PDEs assembled from eight terms on a periodic domain, with each field integrated over a 64 by 64 grid for 50 frames.
- 3.1 Data and benchmark: Accuracy is measured with mean F1 for term-set overlap and exact recovery for exact matches, while the fixed-term set floor quantifies inflation from common-term guesses.
- 3.1 Data and benchmark: The benchmark varies PDE temporal order and nonlinear-term content across six strata, including deliberately unfavorable cases with negligible or difficult-to-distinguish terms.
- 3.2 Results and discussion: Data interpretation exceeds the optimized fixed-term floor by 0.329 at paired Wilcoxon p = 2.6 × 10−5 and doubles exact recovery at p = 6.6 × 10−3.
- 3.2 Results and discussion: Data interpretation reaches F1 = 0.720 versus 0.225 for raw field slices, with exact recovery on 14 of 44 samples versus 2.
- 3.2 Results and discussion: Accuracy is highest where interpretation is most complete: linear-only samples are recovered perfectly, while nonlinear cases remain limited by unknown nonlinear-term form.
4 Concluding remarks
The paper concludes that direct physical measurements let an LLM identify governing equations better than raw fields, supporting a compact, training-free route to automated field theory construction. Its scope is limited by small, noise-free simulated datasets and characterization tools that do not fully specify nonlinear forms and are fragile to noise.
- Limits: The study uses 44 samples, one frozen model, and noise-free simulated fields drawn from a library of eight terms.
- Limits: Current characterizations establish that a nonlinear term is present without indicating its form, reducing identification accuracy for those terms.
- Limits: The reported characterizations are fragile to noise, so robust differentiation and noise filtering are required before application to experimental data.
- 4 Concluding remarks: Data interpretation gives the LLM physical measurements directly, and the resulting equation identification is substantially better than when the model receives the field itself.
- 4 Concluding remarks: The measurements cost a fraction of a second per field, use fewer tokens than the raw data they replace, and require no training.
- 4 Concluding remarks: The pipeline assigns data interpretation to identifying mechanisms and evaluation to testing candidate equations against the data.
- Future work: Future work proposes interpreting equation residuals so the LLM can identify missing dynamics and iteratively revise its proposals.