Moving beyond traditional black-box AI methodologies, what scientists today require is not just simple result prediction, but physics-based AI that can explain “why” a particular result was produced. While this perspective has always been valued, it is only recently that concrete technological developments have begun to fully support it.
In this context, one of the most notable techniques is SISSO (Symbolic Regression with SIS-SO), which we introduced in a previous post. SISSO was an early pioneer of this approach, aiming for “interpretable AI” by providing concise physical formulas that humans can understand. This allows it to offer powerful insights even in experimental environments where data is scarce.
However, there is another compelling perspective to consider:
“While discovering formulas from data (SISSO) is innovative, can’t we maximize learning efficiency by directly teaching AI the thousands of physics laws humanity has already accumulated?”
The recently emerged Physics-Informed Neural Networks (PINNs) are known for breaking the limitations of traditional deep learning—which relies solely on data volume—by directly embedding physical laws into the neural network itself. In this post, we will introduce PINNs as a new milestone in interpretable methodology for AI-driven scientific research, explore how it differs from SISSO, and discuss the synergy created when these two techniques are combined.
Data vs. Physics Laws: Two Distinct Strategies for Teaching AI
Utilizing AI in materials science is ultimately the process of defining correlations between numerous variables. Researchers face two distinct choices: Should they discover unknown rules hidden within the data, or should they inject proven physical principles into the AI? The former is the path of SISSO, and the latter is the path of PINNs. This choice can fundamentally change both the efficiency of the research and the reliability of the resulting data.
SISSO: A Mining Operation to Find Formulas Hidden in Data
The core of SISSO lies in discovery from data. Its essence is to combine numerous physical variables to find the most meaningful correlations and extract them into concise symbolic expressions that researchers can understand at a glance.
- Deriving Interpretable Descriptors: Unlike Principal Component Analysis (PCA), which simply compresses data, SISSO generates descriptors (in formulaic form) that consider the dimensions and units of existing physical laws, allowing humans to assign physical meaning. Instead of the opaque weights of a complex neural network, it provides clear causal relationships in the form of equations, such as “the product of A and B determines the strength of the material.”
- Gaining Insights through Compressed Sensing: Even without prior knowledge, SISSO utilizes Compressed Sensing techniques to select only the most critical terms from a vast pool of candidate mathematical expressions. This process is akin to a mining operation, identifying high-purity gold veins within a massive mine.
- Limitations: However, SISSO may struggle to find valid equations if the experimental data is heavily noisy or if the system is an extremely complex dynamic system that is difficult to define with a single, concise formula.
PINNs: Internalizing Physical Laws
PINNs (Physics-Informed Neural Networks) consider both the flow of data and compliance with physical laws simultaneously. From the very beginning of the learning process, it imposes a strong guideline on the neural network: “This phenomenon must follow thermodynamic differential equations or the law of conservation of energy.”
The core of this process lies in embedding physical laws directly into the Loss Function. PINNs do not just minimize the numerical difference between predicted and experimental values; they verify in real-time whether the results remain mathematically consistent when substituted back into physical formulas. If the AI produces a result that is physically impossible—such as defying gravity or creating energy from nothing—the loss function immediately treats it as a “wrong answer” (specifically, by increasing the residual of the physical equation), forcing the model to self-correct toward physical plausibility.
This approach differs fundamentally from SISSO. While SISSO is a discovery tool that extracts highly explanatory physical equations from disordered data, PINNs act more as a tool that uses proven physical laws as a filter for the neural network. Thanks to this characteristic, PINNs demonstrate relatively strong performance in extrapolation—predicting outcomes in regions where no data exists. While SISSO relies on Compressed Sensing to strengthen patterns in data-sparse situations, PINNs follow the “proven map” of physical laws. This makes PINNs a decisive solution in research fields exploring unknown territories or designing materials for extreme environments where experimental data is difficult to obtain.
Harmonizing Discovery and Compliance
While both SISSO and PINNs share the common goal of integrating physical insight into AI methodologies, their methodological orientations are not identical. The choice of tool depends on whether the research goal is to discover new physical correlations or to perform precise predictions within the boundaries of known laws. Understanding this fundamental difference is the first step toward maximizing research efficiency.
The differences between the two techniques in terms of purpose, input, and output are as follows:
- Purpose:SISSO focuses on the discovery of new physical laws or correlations hidden in data, whereas PINNs focus on complying with given physical laws to predict and optimize system behavior.
- Input:SISSO uses pure experimental (or simulation) data as its primary raw material to find patterns. In contrast, PINNs explicitly receive physical constraints (such as differential equations) as input along with the experimental data.
- Output: The goal of SISSO is to produce concise physical formulas that researchers can directly cite in papers. PINNs, on the other hand, provide a sophisticated predictive model itself that accurately reproduces complex phenomena while strictly adhering to physical laws.
To put it into perspective for materials science, SISSO is like a theorist who discovers a new correlation formula between unknown alloy compositions and mechanical strength. In contrast, PINNs act as a process design expert who precisely simulates manufacturing conditions to ensure that the discovered formula holds true in real-world environments, ultimately finding the optimal manufacturing parameters.
Beyond simply choosing one technique over the other, researchers should consider linking them organically. From a technical and mechanical standpoint, a hybrid strategy of “discovery via SISSO and verification via PINNs” is a highly feasible approach that goes beyond a simple combination of tools.
If researchers successfully identify the governing equations or key descriptors of a new material system using SISSO, they can then inject these discovered formulas into the PINNs loss function. By doing so, PINNs will no longer just follow generic, pre-existing physical laws; instead, they will operate under a customized physical guideline tailored specifically to that material system, derived directly from the researcher’s own data. To ensure scientific rigor and gain trust from the community, this process can be validated by building an iterative loop where experts evaluate the results and feed them back as input data for SISSO, continuously refining the model.
Summary: Complementary Application and Technical Synergy
In this post, we explored the shift in data-driven materials science from black-box AI to “white-box” techniques grounded in physical evidence. We examined how SISSO and PINNs overcome the inherent limitations of data-driven research through their respective strategies of law discovery and law compliance.
While SISSO provides physical insight by uncovering concise formulas hidden within data, PINNs perform physically plausible and precise predictions by embedding physical laws, such as differential equations, directly into neural networks. These two methodologies share a deeply complementary relationship with significant potential for integration. We look forward to seeing how the strategic utilization of these techniques will create powerful technical synergies in the future of materials research.