Andrej Novak

Hasan Akgul, Daniel Borg, Arta Berisha et al. · mila

Large language models are often adapted through parameter efficient fine tuning, but current release practices provide weak assurances about what data were used and how updates were computed. We present Verifiable Fine Tuning, a protocol and system that produces succinct zero knowledge proofs that a released model was obtained from a public initialization under a declared training program and an auditable dataset commitment. The approach combines five elements. First, commitments that bind data sources, preprocessing, licenses, and per epoch quota counters to a manifest. Second, a verifiable sampler that supports public replayable and private index hiding batch selection. Third, update circuits restricted to parameter efficient fine tuning that enforce AdamW style optimizer semantics and proof friendly approximations with explicit error budgets. Fourth, recursive aggregation that folds per step proofs into per epoch and end to end certificates with millisecond verification. Fifth, provenance binding and optional trusted execution property cards that attest code identity and constants. On English and bilingual instruction mixtures, the method maintains utility within tight budgets while achieving practical proof performance. Policy quotas are enforced with zero violations, and private sampling windows show no measurable index leakage. Federated experiments demonstrate that the system composes with probabilistic audits and bandwidth constraints. These results indicate that end to end verifiable fine tuning is feasible today for real parameter efficient pipelines, closing a critical trust gap for regulated and decentralized deployments.

Maria Deliyianni, Boris Muha, Andrej Novak

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field feeds back on the flow through concentration-dependent density and viscosity, yielding a phenomenological temperature-coupled Cahn-Hilliard-Stokes-Heat system. We motivate the chemical potential by a temperature-dependent Landau free energy and derive a priori estimates for the regularized subproblems. On the analytical side, we prove local-in-time existence of weak solutions for a regularized coupled system. On the numerical side, we propose a fully discrete finite element scheme combining a convex-splitting time discretization for the Cahn-Hilliard equation with an implicit treatment of viscous and thermal diffusion terms and a an implicit Stokes solve. Under impermeable velocity boundary conditions, the Cahn-Hilliard substep conserves mass, in the purely diffusive isothermal case, the convex-splitting discretization is unconditionally energy-stable for the Cahn-Hilliard free energy. Numerical experiments in two dimensions illustrate thermally driven spinodal decomposition, wall-induced phase separation near cooled walls, and phase separation in narrow channels under imposed thermal gradients. The simulations show the qualitative influence of key nondimensional parameters (such as the mass and thermal Péclet numbers, the Cahn number, the density and viscosity ratios, and the gravitational parameter $G$) on pattern formation, interface motion, and flow structure, and confirm that the proposed framework is a robust tool for studying thermally driven phase separation in confined geometries.

Darko Mitrovic, Andrej Novak

We extend Brenier's transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws and initial-boundary value problem for homogeneous scalar conservation laws. It is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. In the case of initial-boundary value problem, we such a procedure is used to construct a numerical scheme which leads us to a new solution concept for initial-boundary value problem for scalar conservation laws. The concept is a generalization (refinement) of the previous works on initial-boundary value problem. We also provide numerical examples.