Md Nasim

Md Nasim, Anter El-Azab, Xinghang Zhang et al.

The availability of tera-byte scale experiment data calls for AI driven approaches which automatically discover scientific models from data. Nonetheless, significant challenges present in AI-driven scientific discovery: (i) The annotation of large scale datasets requires fundamental re-thinking in developing scalable crowdsourcing tools. (ii) The learning of scientific models from data calls for innovations beyond black-box neural nets. (iii) Novel visualization and diagnosis tools are needed for the collaboration of experimental and theoretical physicists, and computer scientists. We present Phase-Field-Lab platform for end-to-end phase field model discovery, which automatically discovers phase field physics models from experiment data, integrating experimentation, crowdsourcing, simulation and learning. Phase-Field-Lab combines (i) a streamlined annotation tool which reduces the annotation time (by ~50-75%), while increasing annotation accuracy compared to baseline; (ii) an end-to-end neural model which automatically learns phase field models from data by embedding phase field simulation and existing domain knowledge into learning; and (iii) novel interfaces and visualizations to integrate our platform into the scientific discovery cycle of domain scientists. Our platform is deployed in the analysis of nano-structure evolution in materials under extreme conditions (high temperature and irradiation). Our approach reveals new properties of nano-void defects, which otherwise cannot be detected via manual analysis.

Nan Jiang, Md Nasim, Yexiang Xue

The symbolic discovery of Ordinary Differential Equations (ODEs) from trajectory data plays a pivotal role in AI-driven scientific discovery. Existing symbolic methods predominantly rely on fixed, pre-collected training datasets, which often result in suboptimal performance, as demonstrated in our case study in Figure 1. Drawing inspiration from active learning, we investigate strategies to query informative trajectory data that can enhance the evaluation of predicted ODEs. However, the butterfly effect in dynamical systems reveals that small variations in initial conditions can lead to drastically different trajectories, necessitating the storage of vast quantities of trajectory data using conventional active learning. To address this, we introduce Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching (APPS). Instead of directly selecting individual initial conditions, our APPS first identifies an informative region within the phase space and then samples a batch of initial conditions from this region. Compared to traditional active learning methods, APPS mitigates the gap of maintaining a large amount of data. Extensive experiments demonstrate that APPS consistently discovers more accurate ODE expressions than baseline methods using passively collected datasets.

Md Nasim, Yexiang Xue

Accelerating the learning of Partial Differential Equations (PDEs) from experimental data will speed up the pace of scientific discovery. Previous randomized algorithms exploit sparsity in PDE updates for acceleration. However such methods are applicable to a limited class of decomposable PDEs, which have sparse features in the value domain. We propose Reel, which accelerates the learning of PDEs via random projection and has much broader applicability. Reel exploits the sparsity by decomposing dense updates into sparse ones in both the value and frequency domains. This decomposition enables efficient learning when the source of the updates consists of gradually changing terms across large areas (sparse in the frequency domain) in addition to a few rapid updates concentrated in a small set of "interfacial" regions (sparse in the value domain). Random projection is then applied to compress the sparse signals for learning. To expand the model applicability, Taylor series expansion is used in Reel to approximate the nonlinear PDE updates with polynomials in the decomposable form. Theoretically, we derive a constant factor approximation between the projected loss function and the original one with poly-logarithmic number of projected dimensions. Experimentally, we provide empirical evidence that our proposed Reel can lead to faster learning of PDE models (70-98% reduction in training time when the data is compressed to 1% of its original size) with comparable quality as the non-compressed models.

Nan Jiang, Md Nasim, Yexiang Xue

Vertical Symbolic Regression (VSR) recently has been proposed to expedite the discovery of symbolic equations with many independent variables from experimental data. VSR reduces the search spaces following the vertical discovery path by building from reduced-form equations involving a subset of independent variables to full-fledged ones. Proved successful by many symbolic regressors, deep neural networks are expected to further scale up VSR. Nevertheless, directly combining VSR with deep neural networks will result in difficulty in passing gradients and other engineering issues. We propose Vertical Symbolic Regression using Deep Policy Gradient (VSR-DPG) and demonstrate that VSR-DPG can recover ground-truth equations involving multiple input variables, significantly beyond both deep reinforcement learning-based approaches and previous VSR variants. Our VSR-DPG models symbolic regression as a sequential decision-making process, in which equations are built from repeated applications of grammar rules. The integrated deep model is trained to maximize a policy gradient objective. Experimental results demonstrate that our VSR-DPG significantly outperforms popular baselines in identifying both algebraic equations and ordinary differential equations on a series of benchmarks.

Nan Jiang, Md Nasim, Yexiang Xue

Automating scientific discovery has been a grand goal of Artificial Intelligence (AI) and will bring tremendous societal impact. Learning symbolic expressions from experimental data is a vital step in AI-driven scientific discovery. Despite exciting progress, most endeavors have focused on the horizontal discovery paths, i.e., they directly search for the best expression in the full hypothesis space involving all the independent variables. Horizontal paths are challenging due to the exponentially large hypothesis space involving all the independent variables. We propose Vertical Symbolic Regression (VSR) to expedite symbolic regression. The VSR starts by fitting simple expressions involving a few independent variables under controlled experiments where the remaining variables are held constant. It then extends the expressions learned in previous rounds by adding new independent variables and using new control variable experiments allowing these variables to vary. The first few steps in vertical discovery are significantly cheaper than the horizontal path, as their search is in reduced hypothesis spaces involving a small set of variables. As a consequence, vertical discovery has the potential to supercharge state-of-the-art symbolic regression approaches in handling complex equations with many contributing factors. Theoretically, we show that the search space of VSR can be exponentially smaller than that of horizontal approaches when learning a class of expressions. Experimentally, VSR outperforms several baselines in learning symbolic expressions involving many independent variables.

Joshua Fan, Haodi Xu, Feng Tao et al.

Understanding how carbon flows through the soil is crucial for mitigating the effects of climate change. While soils have potential to sequester carbon from the atmosphere, the soil carbon cycle remains poorly understood. Scientists have developed mathematical process-based models of the soil carbon cycle based on existing knowledge, but they contain numerous unknown parameters that must be set in an ad-hoc manner, and often fit observations poorly. On the other hand, neural networks can learn patterns from data, but do not respect known scientific laws, nor can they reveal novel scientific relationships due to their black-box nature. We thus propose Scientifically-Interpretable Reasoning Network (ScIReN), a fully-transparent framework that combines interpretable neural and process-based reasoning. An interpretable encoder predicts scientifically-meaningful latent parameters, which are then passed through a differentiable process-based decoder to predict labeled output variables. ScIReN leverages Kolmogorov-Arnold networks (KAN) to ensure the encoder is fully interpretable and reveals relationships between input features and latent parameters; it uses novel smoothness penalties to balance expressivity and simplicity. ScIReN also uses a novel hard-sigmoid constraint layer to restrict latent parameters to meaningful ranges defined by scientific prior knowledge. While the process-based decoder enforces established scientific knowledge, the KAN-based encoder reveals new scientific relationships hidden in conventional black-box models. We apply ScIReN on two tasks: simulating the flow of organic carbon through soils, and modeling ecosystem respiration from plants. In both tasks, ScIReN outperforms black-box networks in predictive accuracy while providing substantial scientific interpretability -- it can infer latent scientific mechanisms and their relationships with input features.