Mathematical foundations for higher-order scientific laws — where pairwise graphs fall short, a $1M award (UC Santa Barbara)
Many physical systems, from molecules to galaxies, are driven by interactions among their components, which AI for science typically models with graphs and graph neural networks. But physical systems also involve richer higher-order interactions of varying strengths beyond the pairwise relations graphs handle. This project defines the mathematical foundations of higher-order extensions.
Grant overview (primary data)
- Award amount$1,000,000
- RecipientUniversity of California-Santa Barbara (California)
- ProgramSpecial Projects - CCF, Special Initiatives, MSPA-INTERDISCIPLINARY, EPCL: Energy, Power, Control,
- Period2026-09-01 〜 2029-08-31
- FunderU.S. National Science Foundation (NSF) / NSF
Key points
- Many physical systems from molecules to galaxies are driven by interactions among components, modeled in AI for science with graphs and graph neural networks.
- Graphs assume pairwise relations and therefore miss higher-order structures such as carbon rings, functional groups and weak non-covalent forces.
- The record notes that in some fields the physical laws governing higher-order interactions remain undiscovered, not merely unrepresented.
- The main difficulty is the combinatorial explosion inherent in studying higher-order interactions.
- The project defines the mathematical foundations of higher-order extensions of graph neural networks. Estimated total and obligated amount are both $1,000,000.
- Drawing a molecule as a graph drops structures meaningful only as a whole, such as a carbon ring, and weak non-covalent forces.
1The limit of describing a world in pairs
A graph expresses relations by joining points with lines: atom to atom, brain region to brain region, star to star. Easy to handle, it also assumes relations always exist between two things. The example the record gives is clear.
Represent a molecule as a graph and atoms become points and covalent bonds lines, while structures meaningful only when several atoms come together — a carbon ring — along with functional groups and weak non-covalent forces, fall out of that representation. The frame of description decides what can be seen.
2Fields where the laws themselves are unknown
The record goes a step further, noting that in some fields not only are higher-order interactions unrepresented but the physical laws governing them remain undiscovered. It is not that something cannot be handled because it cannot be expressed; what should be expressed is not yet known. Bringing AI into this territory aims not at computing known laws faster but at finding which interactions matter, at which scale.
3The wall of combinatorial explosion
The main difficulty named is combinatorial explosion. With pairwise relations the number of combinations stays limited, but raising it to three or four makes candidates grow rapidly until exhaustive examination becomes impossible. Building a framework for higher-order interactions is the same problem as containing that explosion.
The 120 NSF awards this site holds as of 2026-08-31 span 67 programs, and this one belongs to a framework addressing mathematical foundations — investment in the mathematics that sits before application.
4What drops out the moment you draw a graph
Turning a molecule into a graph is a choice about what to keep and what to discard. Atoms become nodes and covalent bonds become edges. That much transfers, but a structure that means something only when several atoms act together cannot be written as a set of nodes and edges.
Building a framework for the right column is this project's problem. And in some fields the physical laws governing that column have not been found at all. The limit of the representation has become the limit of what can be seen.
Why it matters
Applying AI to science divides between computing known things faster and finding structures not yet known. This project is the latter, investing in the mathematics that sits before application. That a frame of description decides what can be seen carries beyond science, since how data is held sets the limits of analysis.
FAQ
What are higher-order interactions?
Why are graphs insufficient?
What is combinatorial explosion?
Sources (primary)
Source: NSF Award Search (U.S. National Science Foundation, public domain). Amounts are the obligated amount. For privacy, we do not handle principal investigator names.
- NSF Award (original, official)
- NSF Award ID: 2602079