Formalizing Inverse Problems in the Age of AI: Possibilities, Pitfalls, and an Invitation
Authors: Christian Clason, Tapio Helin, Medet Nursultanov, Lauri Oksanen
arXiv: 2611.06962 [cs.AI]
Submitted: 4 November 2026
MSC Classes: 68V20, 47A52
Abstract
Proof assistants such as Lean check mathematical proofs down to the axioms, and AI agents have recently made writing such proofs inexpensive. This paper describes AutoIP, a Lean library in which classical linear regularization theory—together with the functional analysis on which it rests—has been autoformalized by AI agents.
A machine-checked proof certifies only the formal statement it proves. The remaining human task is therefore to confirm that each formal statement says what was intended. The library is organized around this task: a reviewer compares formal statements with their natural-language counterparts and never needs to read a proof. The authors' experience indicates that the risks lie in the statements rather than in the proofs, and that simple automated tests on statements can expose problems even in the established literature. This is illustrated with a textbook theorem whose hypothesis is more restrictive than asserted.
The authors invite the inverse problems community to extend and build on the library.
Key Contributions
- AutoIP: A Lean library autoformalizing classical linear regularization theory and its underlying functional analysis.
- Reviewer-centered design: Human effort focuses on validating formal statements against their natural-language counterparts, not on verifying proof steps.
- Statement-level risk: Empirical observation that errors concentrate in formalized statements rather than proofs.
- Automated statement testing: Simple automated checks can reveal issues in established results, demonstrated via a textbook theorem with an over-general hypothesis.
- Community invitation: An open call for the inverse problems community to extend and build on the library.
Context (2026)
By 2026, AI agents have substantially lowered the cost of producing machine-checkable proofs in systems such as Lean, shifting the bottleneck in formalization from proof construction to statement fidelity. This work aligns with a broader trend toward autoformalization pipelines in which AI generates formal artifacts and humans audit specifications, a division of labor increasingly central to trustworthy mathematical and scientific computing.
Citation
Clason, C., Helin, T., Nursultanov, M., & Oksanen, L. (2026).
Formalizing inverse problems in the age of AI: possibilities, pitfalls, and an invitation.
arXiv:2611.06962 [cs.AI]. https://doi.org/10.48550/arXiv.2611.06962
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