Kazem Meidani

Kazem Meidani, Parshin Shojaee, Chandan K. Reddy et al.

In an era where symbolic mathematical equations are indispensable for modeling complex natural phenomena, scientific inquiry often involves collecting observations and translating them into mathematical expressions. Recently, deep learning has emerged as a powerful tool for extracting insights from data. However, existing models typically specialize in either numeric or symbolic domains, and are usually trained in a supervised manner tailored to specific tasks. This approach neglects the substantial benefits that could arise from a task-agnostic multi-modal understanding between symbolic equations and their numeric counterparts. To bridge the gap, we introduce SNIP, a Symbolic-Numeric Integrated Pre-training model, which employs contrastive learning between symbolic and numeric domains, enhancing their mutual similarities in the embeddings. By performing latent space analysis, we observe that SNIP provides cross-domain insights into the representations, revealing that symbolic supervision enhances the embeddings of numeric data and vice versa. We evaluate SNIP across diverse tasks, including symbolic-to-numeric mathematical property prediction and numeric-to-symbolic equation discovery, commonly known as symbolic regression. Results show that SNIP effectively transfers to various tasks, consistently outperforming fully supervised baselines and competing strongly with established task-specific methods, especially in the low data regime scenarios where available data is limited. Code and model are available at: https://github.com/deep-symbolic-mathematics/Multimodal-Math-Pretraining

Zijie Li, Kazem Meidani, Amir Barati Farimani

Data-driven learning of partial differential equations' solution operators has recently emerged as a promising paradigm for approximating the underlying solutions. The solution operators are usually parameterized by deep learning models that are built upon problem-specific inductive biases. An example is a convolutional or a graph neural network that exploits the local grid structure where functions' values are sampled. The attention mechanism, on the other hand, provides a flexible way to implicitly exploit the patterns within inputs, and furthermore, relationship between arbitrary query locations and inputs. In this work, we present an attention-based framework for data-driven operator learning, which we term Operator Transformer (OFormer). Our framework is built upon self-attention, cross-attention, and a set of point-wise multilayer perceptrons (MLPs), and thus it makes few assumptions on the sampling pattern of the input function or query locations. We show that the proposed framework is competitive on standard benchmark problems and can flexibly be adapted to randomly sampled input.

Parshin Shojaee, Kazem Meidani, Amir Barati Farimani et al.

Symbolic regression (SR) is a challenging task in machine learning that involves finding a mathematical expression for a function based on its values. Recent advancements in SR have demonstrated the effectiveness of pre-trained transformer-based models in generating equations as sequences, leveraging large-scale pre-training on synthetic datasets and offering notable advantages in terms of inference time over classical Genetic Programming (GP) methods. However, these models primarily rely on supervised pre-training goals borrowed from text generation and overlook equation discovery objectives like accuracy and complexity. To address this, we propose TPSR, a Transformer-based Planning strategy for Symbolic Regression that incorporates Monte Carlo Tree Search into the transformer decoding process. Unlike conventional decoding strategies, TPSR enables the integration of non-differentiable feedback, such as fitting accuracy and complexity, as external sources of knowledge into the transformer-based equation generation process. Extensive experiments on various datasets show that our approach outperforms state-of-the-art methods, enhancing the model's fitting-complexity trade-off, extrapolation abilities, and robustness to noise.

Parshin Shojaee, Kazem Meidani, Shashank Gupta et al.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely large combinatorial hypothesis spaces. Current methods of equation discovery, commonly known as symbolic regression techniques, largely focus on extracting equations from data alone, often neglecting the domain-specific prior knowledge that scientists typically depend on. They also employ limited representations such as expression trees, constraining the search space and expressiveness of equations. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeleton hypotheses, drawing from its domain knowledge, which are then optimized against data to estimate parameters. We evaluate LLM-SR on four benchmark problems across diverse scientific domains (e.g., physics, biology), which we carefully designed to simulate the discovery process and prevent LLM recitation. Our results demonstrate that LLM-SR discovers physically accurate equations that significantly outperform state-of-the-art symbolic regression baselines, particularly in out-of-domain test settings. We also show that LLM-SR's incorporation of scientific priors enables more efficient equation space exploration than the baselines. Code and data are available: https://github.com/deep-symbolic-mathematics/LLM-SR

Parshin Shojaee, Ngoc-Hieu Nguyen, Kazem Meidani et al.

Scientific equation discovery is a fundamental task in the history of scientific progress, enabling the derivation of laws governing natural phenomena. Recently, Large Language Models (LLMs) have gained interest for this task due to their potential to leverage embedded scientific knowledge for hypothesis generation. However, evaluating the true discovery capabilities of these methods remains challenging, as existing benchmarks often rely on common equations that are susceptible to memorization by LLMs, leading to inflated performance metrics that do not reflect discovery. In this paper, we introduce LLM-SRBench, a comprehensive benchmark with 239 challenging problems across four scientific domains specifically designed to evaluate LLM-based scientific equation discovery methods while preventing trivial memorization. Our benchmark comprises two main categories: LSR-Transform, which transforms common physical models into less common mathematical representations to test reasoning beyond memorized forms, and LSR-Synth, which introduces synthetic, discovery-driven problems requiring data-driven reasoning. Through extensive evaluation of several state-of-the-art methods, using both open and closed LLMs, we find that the best-performing system so far achieves only 31.5% symbolic accuracy. These findings highlight the challenges of scientific equation discovery, positioning LLM-SRBench as a valuable resource for future research.

Mayuka Jayawardhana, Nihal Sharma, Kazem Meidani et al.

Tabular foundation models, particularly Prior-data Fitted Networks like TabPFN have emerged as the leading contender in a myriad of tasks ranging from data imputation to label prediction on the tabular data format surpassing the historical successes of tree-based models. This has led to investigations on their applicability to forecasting time series data which can be formulated as a tabular problem. While recent work to this end has displayed positive results, most works have limited their treatment of multivariate time series problems to several independent univariate time series forecasting subproblems, thus ignoring any inter-channel interactions. Overcoming this limitation, we introduce a generally applicable framework for multivariate time series forecasting using tabular foundation models. We achieve this by recasting the multivariate time series forecasting problem as a series of scalar regression problems which can then be solved zero-shot by any tabular foundation model with regression capabilities. We present results of our method using the TabPFN-TS backbone and compare performance with the current state of the art tabular methods.

Benjamin Léger, Kazem Meidani, Christian Gagné

Symbolic regression (SR) aims to discover mathematical expressions from data, a task traditionally tackled using Genetic Programming (GP) through combinatorial search over symbolic structures. Latent Space Optimization (LSO) methods use neural encoders to map symbolic expressions into continuous spaces, transforming the combinatorial search into continuous optimization. SNIP (Meidani et al., 2024), a contrastive pre-training model inspired by CLIP, advances LSO by introducing a multi-modal approach: aligning symbolic and numeric encoders in a shared latent space to learn the phenotype-genotype mapping, enabling optimization in the numeric space to implicitly guide symbolic search. However, this relies on fine-grained cross-modal alignment, whereas literature on similar models like CLIP reveals that such an alignment is typically coarse-grained. In this paper, we investigate whether SNIP delivers on its promise of effective bi-modal optimization for SR. Our experiments show that: (1) cross-modal alignment does not improve during optimization, even as fitness increases, and (2) the alignment learned by SNIP is too coarse to efficiently conduct principled search in the symbolic space. These findings reveal that while multi-modal LSO holds significant potential for SR, effective alignment-guided optimization remains unrealized in practice, highlighting fine-grained alignment as a critical direction for future work.

Parand Akbari, Francis Ogoke, Ning-Yu Kao et al.

Characterizing meltpool shape and geometry is essential in metal Additive Manufacturing (MAM) to control the printing process and avoid defects. Predicting meltpool flaws based on process parameters and powder material is difficult due to the complex nature of MAM process. Machine learning (ML) techniques can be useful in connecting process parameters to the type of flaws in the meltpool. In this work, we introduced a comprehensive framework for benchmarking ML for melt pool characterization. An extensive experimental dataset has been collected from more than 80 MAM articles containing MAM processing conditions, materials, meltpool dimensions, meltpool modes and flaw types. We introduced physics-aware MAM featurization, versatile ML models, and evaluation metrics to create a comprehensive learning framework for meltpool defect and geometry prediction. This benchmark can serve as a basis for melt pool control and process optimization. In addition, data-driven explicit models have been identified to estimate meltpool geometry from process parameters and material properties which outperform Rosenthal estimation for meltpool geometry while maintaining interpretability.

Zijie Li, Kazem Meidani, Prakarsh Yadav et al.

Molecular Dynamics (MD) simulation is a powerful tool for understanding the dynamics and structure of matter. Since the resolution of MD is atomic-scale, achieving long time-scale simulations with femtosecond integration is very expensive. In each MD step, numerous iterative computations are performed to calculate energy based on different types of interaction and their corresponding spatial gradients. These repetitive computations can be learned and surrogated by a deep learning model like a Graph Neural Network (GNN). In this work, we developed a GNN Accelerated Molecular Dynamics (GAMD) model that directly predicts forces given the state of the system (atom positions, atom types), bypassing the evaluation of potential energy. By training the GNN on a variety of data sources (simulation data derived from classical MD and density functional theory), we show that GAMD can predict the dynamics of two typical molecular systems, Lennard-Jones system and Water system, in the NVT ensemble with velocities regulated by a thermostat. We further show that GAMD's learning and inference are agnostic to the scale, where it can scale to much larger systems at test time. We also perform a comprehensive benchmark test comparing our implementation of GAMD to production-level MD softwares, showing GAMD's competitive performance on the large-scale simulation.

Francis Ogoke, Kazem Meidani, Amirreza Hashemi et al.

Many scientific and engineering processes produce spatially unstructured data. However, most data-driven models require a feature matrix that enforces both a set number and order of features for each sample. They thus cannot be easily constructed for an unstructured dataset. Therefore, a graph based data-driven model to perform inference on fields defined on an unstructured mesh, using a Graph Convolutional Neural Network (GCNN) is presented. The ability of the method to predict global properties from spatially irregular measurements with high accuracy is demonstrated by predicting the drag force associated with laminar flow around airfoils from scattered velocity measurements. The network can infer from field samples at different resolutions, and is invariant to the order in which the measurements within each sample are presented. The GCNN method, using inductive convolutional layers and adaptive pooling, is able to predict this quantity with a validation $R^{2}$ above 0.98, and a Normalized Mean Squared Error below 0.01, without relying on spatial structure.

Kazem Meidani, Amir Barati Farimani

Many scientific phenomena are modeled by Partial Differential Equations (PDEs). The development of data gathering tools along with the advances in machine learning (ML) techniques have raised opportunities for data-driven identification of governing equations from experimentally observed data. We propose an ML method to discover the terms involved in the equation from two-dimensional spatiotemporal data. Robust and useful physical features are extracted from data samples to represent the specific behaviors imposed by each mathematical term in the equation. Compared to the previous models, this idea provides us with the ability to discover 2D equations with time derivatives of different orders, and also to identify new underlying physics on which the model has not been trained. Moreover, the model can work with small sets of low-resolution data while avoiding numerical differentiations. The results indicate robustness of the features extracted based on prior knowledge in comparison to automatically detected features by a Three-dimensional Convolutional Neural Network (3D CNN) given the same amounts of data. Although particular PDEs are studied in this work, the idea of the proposed approach could be extended for reliable identification of various PDEs.