数学前沿

OpenAI publishes new results on open problems in mathematics from an internal frontier model and shares Lean proof formalizations and research details on GitHub.

OpenAI is working with an independent Advisory Group on Mathematics and Artificial Intelligence to guide the review and communication of emerging AI results.

Acknowledgments of AI use in arXiv math preprints rose from 4% in April 2026 to 25% in August, with 6% crediting AI with a substantial research contribution.

OpenAI's GPT-6 Astra tops the Epoch Capabilities Index (ECI) with a score of 166, ahead of Claude Fable 5.1 at 164 and GPT-5.6 Sol at 162. Its Math-ECI of 170 sets a new record, but on software engineering benchmarks its SWE-ECI of 164 still lags behind Fable 5.1's 167.

We’re sharing an AI-generated solution to the Navier–Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean.

A benchmark of 68 significant Erdős problems, open as of August 2026, curated by Thomas Bloom and formalized in Lean. AI systems must prove or disprove them within a fixed budget.

OpenAI shares new results on long-standing open problems in mathematics and theoretical computer science, including advances in geometry, cryptography, and complexity.

Research papers point to the growing impact of Deep Think across fields

UCLA Professor Ernest Ryu and GPT-5 solved a key question in optimization theory, showcasing AI’s role in accelerating mathematical discovery.

We’ve trained a model to achieve a new state-of-the-art in mathematical problem solving by rewarding each correct step of reasoning (“process supervision”) instead of simply rewarding the correct final answer (“outcome supervision”). In addition to boosting performance relative to outcome supervision, process supervision also has an important alignment benefit: it directly trains the model to produce a chain-of-thought that is endorsed by humans.

We built a neural theorem prover for Lean that learned to solve a variety of challenging high-school olympiad problems, including problems from the AMC12 and AIME competitions, as well as two problems adapted from the IMO.

Symmetric polyconvexity provides a sufficient condition for weak lower semicontinuity in small-strain elasticity and a tractable lower bound for the relaxation of non-(quasi)convex energies. Since classical algorithms for computing such functions can be computationally expensive, neural network approximations have recently emerged as an alternative. A central difficulty with establishing such neural network based approaches for symmetric polyconvexity is the additional monotonicity property. As we show, the corresponding weight restrictions on ReLU-based input-convex networks do not yield universal approximation of the lifted representers. Here, we introduce symmetric polyconvex neural networks (SPCNNs) for constitutive modeling and approximation of symmetric polyconvex envelopes. The deep architecture combines convexity with the monotonicity required of a representer in the space of minors. We prove a universal approximation theorem for symmetric polyconvex energies, without requiring isotropy, and quantify the number of affine pieces attainable by the lifted network and derive reduced architectures for isotropic energies. Numerical examples illustrate stress-based regression and approximation of the symmetric polyconvex envelope of a double-well potential. Symmetric polyconvexity is guaranteed by construction, while the accuracy of the fitted energies and envelopes is assessed numerically.

Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within a probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric---both in the design of the loss function and in the optimization process---significantly impacts the stability and properties of the optimizer.

Contextual decision making uses observed covariates to guide decisions under uncertainty. Two natural approaches are predict-then-optimize (PtO), which learns the conditional expected objective, and generate-then-optimize (GtO), which learns conditional uncertainty and optimizes using generated scenarios. We study when each approach is preferable by examining how the choice of learning target affects statistical accuracy and optimization tractability. Although PtO learns a narrower target, our regret bounds show how differences in smoothness can offset GtO's additional dimensional burden. Exploiting smoothness also raises an optimization tradeoff: within the class of nonnegative-weight PtO methods studied, a lower bound limits the gains from additional smoothness, while more flexible methods may introduce signed weights and lose the guarantee of preserving convexity. GtO preserves scenario-wise convexity, but requires learning a richer object and generating scenarios for optimization. Synthetic experiments and a real case study illustrate how these tradeoffs vary with data availability and problem structure. Together with the scope for reusing a model across tasks, our findings provide practical guidance for choosing between the approaches without identifying a universal winner.

We study the dynamic allocation of scarce medications in Liberia's community health program. A central warehouse supplies a health facility and the Community Health Assistants (CHAs) it supervises. When a CHA runs out of medication, patients are referred to the facility, incurring travel costs if they complete the referral and remain untreated otherwise. Allocating inventory to CHAs allows more patients to receive treatment without traveling, while retaining inventory centrally preserves flexibility to respond to uncertain demand. The allocation problem must balance these considerations while accounting for fairness across locations. We formulate this problem as a finite-horizon stochastic control problem whose objective combines the costs of referrals and untreated patients with a demand-weighted fairness utility. Because standard dynamic programming methods are computationally intractable in this setting, we develop a continuous-time approximation that involves controlling the drift rate of a reflected Brownian motion. We then adapt the deep splitting method, using deep neural networks to approximate the value function and its gradient and compute a dynamic allocation policy. We calibrate the model using data on malaria cases among children in the Fangonda region of Lofa County, Liberia, and evaluate the proposed policy in the original discrete-time model. In the baseline setting, our policy reduces operating costs by about 6% relative to the strongest benchmark, the fairness-only policy, while retaining approximately 98% of its fairness utility. Additional experiments show that higher referral costs shift inventory toward CHAs and reduce local unmet demand, while greater demand variability increases our policy's performance over the fairness-only benchmark. Our model and computational approach provide a framework for studying similar last-mile allocation problems.

A theory can accommodate a counterexample by adding an exception. The harder problem is deciding how far the exception should extend and which predictions should survive. We present DeFAb, a dataset and symbolic verification framework built from curated biological knowledge in the Gene Ontology. Models fill gaps in a theory or propose exceptions to its rules. A verifier checks the consequences of each edit, allowing different solutions to receive credit. Simple symbolic repairs solve all 336 tasks in the released evaluation panel. This result exposes a limitation of the revision test. An exception tied to one entity can succeed without identifying a pattern that applies to other cases. We develop a contextual task that requires an exception to reach the relevant cases and preserve predictions elsewhere. Two planned studies compare training with verified and rejected examples, and revision with and without feedback about consequences. DeFAb makes the scope of a proposed explanation testable, providing a controlled step toward studying theory revision in scientific reasoning.

Stochastic Push-Pull achieves linear speedup with respect to the network size $n$ on static strongly connected digraphs, but whether this speedup persists on time-varying networks remains unclear. In this work, we construct a quadratic decentralized problem on period-two networks showing that stochastic Push-Pull may fail to attain linear speedup over time-varying digraphs. For $p$-periodic networks, we identify an effective node count $n_{\mathrm{eff}}\le n$ that governs the leading stochastic term. Specifically, for smooth nonconvex objectives, we establish convergence upper bounds and matching lower bounds for stochastic Push-Pull with constant stepsizes, both exhibiting the leading-order dependence $\Theta(1/\sqrt{n_{\mathrm{eff}} T})$. When $p=1$, the identity $n_{\mathrm{eff}}=n$ recovers the usual linear speedup for static digraphs. However, periodic networks can have $n_{\mathrm{eff}}=o(n)$, demonstrating the loss of linear speedup under temporal variation. Numerical experiments illustrate effective-count scaling and the separation between static and periodic networks, consistent with our theoretical findings.

Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.