Principia for Scientific Research

We train foundation models from scratch to do original mathematical and scientific reasoning. This is a brief for researchers at national labs and affiliated groups evaluating whether our tools could help with a specific problem.

The technical bet

Most AI-for-science efforts take a general-purpose model like GPT-5, Claude, or Llama and fine-tune it toward a scientific domain. We think this is backwards. Post-training moves under 10% of a model's weights; the other 90% is set during pretraining. If you want a model that reasons mathematically and scientifically, pretraining is where that ability gets built, and fine-tuning a general model shapes only the last mile.

Principia trains base models from scratch with mathematical and scientific reasoning as the objective from weight initialization forward.

Problems we want to work on

This is not exhaustive; it is the texture of what we're set up for. Anywhere researchers are writing custom algorithms, building solvers, or doing heuristic search through a space of mathematical objects, we want to be useful.

Solvers, preconditioners, and numerical schemes. Finding better-performing numerical methods for specific operators: preconditioners for stiff or ill-conditioned systems, discretization schemes with better stability or efficiency properties, algorithm improvements for iterative linear solvers.

Control and reinforcement learning for scientific systems. Improving sample efficiency, robustness, or worst-case guarantees for control methods where experiments are expensive or slow: plasma confinement, grid dispatch, accelerator tuning, robotics, adaptive experimental design.

Symbolic regression and closure modeling. Recovering governing equations, conservation laws, or closure terms from experimental or simulation data. Putting mathematical structure on empirical observations.

Algorithm analysis and formalization of computational workflows. Complexity analysis, scaling bounds, and formal comparison for the ad-hoc pipelines that grow up around real scientific codebases. Making it possible to compare methods rigorously rather than empirically.

Mixed-precision and approximate computation. Extending mixed-precision ideas beyond numerical precision to the structure of the approximation itself. When can a less-rigorous method achieve comparable results, and under what bounds?

Current external engagements include grid optimization, fusion control, and statistical methods for genomics.

What we are not

We are not the right tool for observational sciences without meaningful mathematical structure: field taxonomy, descriptive geology, behavioral ecology. If the hard part of your problem isn't computation, symbol manipulation, or algorithmic reasoning, we probably can't help.

How the model uses tools

The model writes Lean when exactness is required and you want a machine-checkable proof certificate. It writes Python, Julia, and C++ for numerical work, and uses domain-specific languages and solvers where they already exist. The broad middle is physics-style reasoning, approximate arguments, derivations where you can afford to be off by a factor. For those, it reasons in natural language with the flexibility that implies. We pick the right tool for the problem rather than forcing everything through one formalism.

How to work with us

We are not looking for funding or licensing revenue from these engagements. We develop new models on a regular cadence and want their capabilities shaped by problems that matter. The best outcome, for us, is that at the next model release a researcher at a national lab finds the model useful for a problem they care about.

Principia is registered with the Genesis Consortium and happy to engage through that channel or directly.

Operational posture. Researchers retain full IP and publication rights over work produced in these engagements. We can operate under NDA for sensitive or pre-publication material, handle CUI on a case-by-case basis, and are not set up for classified work. For on-premise or air-gapped deployment, contact us and we'll discuss.

A pilot looks like this. Send us a well-posed problem. A paragraph is enough to start. We'll spend about a week on it and come back with either a useful result, a partial result, or an honest account of why we couldn't help. Any of the three outcomes is informative.

Contact

Harry Sanders, harry@principialabs.org. Send a paragraph describing your problem. We respond within two business days.