#01Nov 5, 2026
math.AP
Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision
Michal Shavit, Yongji Wang, Tristan Buckmaster
We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations. We demonstrate the approach for periodic gravity-capillary traveling water waves. Physics-informed neural networks are used as a search mechanism, with the wave profile, wave speed and, in continuation experiments, surface tension learned directly from the governing equations without training data. Through a staged training and refinement procedure, the neural networks achieve near-machine-precision residuals of the governing equations, even in strongly nonlinear regimes. A single formulation recovers high-amplitude Stokes waves, Crapper waves and Wilton ripples. Continuation from a high-amplitude Wilton ripple in surface tension reveals new high-amplitude candidate solutions. We freeze one candidate as an explicit finite representation and rigorously certify a uniform bound on its governing-equation residual using interval arithmetic, accounting for floating-point roundoff, Fourier truncation, aliasing and grid-to-continuum errors. This independently validates the neural-network accuracy. We formalize the certification argument in Lean, which machine-checks the deductions from certified numerical bounds to the residual theorem. Both the interval certification and formalization were developed with substantial assistance from a large language model, while the validity of the result remains independent of the language model: the numerical bounds are rigorously enclosed by interval arithmetic and the subsequent mathematical deductions are checked by the Lean kernel. Together, these steps provide a pipeline from neural discovery to rigorous numerical certification and machine-checked verification with auditable mathematical guarantees.
#02Nov 5, 2026
cs.DB
Selective Retrieval Routing for Multi-Violation Knowledge Graph Repair
Tung-Wei Lin, Gabe Fierro, Han Li and 3 more
Knowledge graphs often contain missing or inconsistent facts that undermine the applications built on them. Schema validation detects such violations but does not identify which of several valid repairs recovers the intended graph. External corpora can supply the evidence needed to identify the intended repair, but retrieving from them is costly and not always successful. We propose a framework for selective retrieval in iterative graph repair. An LLM repair agent explores and edits the graph, while a router controls its access to a researcher agent that retrieves external evidence. We extend a violation-injection framework to generate cases with multiple compatible violations and known reference repairs. The agent's repair outcomes on these cases provide supervision for training the deployment-specific router. On two building-automation datasets, researcher access improves repairs for some violations and degrades them for others, with effects differing across deployments. The router uses fewer repair-time tokens than always enabling access in all six evaluation suites. It achieves higher repair $F_1$ than always disabling access in five of six suites.
#03Nov 5, 2026
cs.LG
Adversarial Multiplicity: Cherry-Picked Models and the Limits of Disclosure
Cassandra C. Chou, Benjamin Q. Huynh, Allison Koenecke and 1 more
Predictive multiplicity -- where similarly accurate models disagree on individual predictions -- can be leveraged to optimize secondary metrics (e.g., fairness) without loss in predictive performance. However, exploiting multiplicity opens the opportunity for motivated parties (or coding agents gaming their objectives) to steer predictions toward their own ends, such as in litigation or consumer disputes with automated decision-making tools. In these adversarial multiplicity settings, we distinguish soft manipulation, where parties optimize over innocuous model choices (e.g., hyperparameter tuning, feature engineering), from hard manipulation, where parties directly perturb model parameters. We find a sufficiently flexible model can be manipulated to reverse any individual prediction with minimal change in predictive performance while passing black-box audits, making hard manipulation viable in most regulatory settings. Soft manipulation has smaller effects but can pass white-box audits requiring full access to code and data. Even aggregate estimates can be manipulated: a model's estimated effect of climate change on wildfires can be fully reversed by hard edits and soft search. We find the Reconcile algorithm typically resolves most disagreements between models, but more often in favor of the manipulator when the opposing model is chosen adversarially compared to at random. We map which manipulations can plausibly pass existing regulations and disclosure policies, and recommend mitigations such as pre-registration, proof of model training, sensitivity analyses, multi-metric evaluations, and committing to versioned public models.
#04Nov 5, 2026
cs.LO
Typed Knowledge over Constructive Institutions
Hans-Martin Will
Type theories designed for mathematics or program verification need only that a proposition holds; proof irrelevance in conversion erases, at the kernel level, how a claim was established. A persistent knowledge store, by contrast, must retain that record: who established a claim, by which derivation or external tool, and under which assumptions. This volume presents the theory of Eigenius---a store of typed knowledge over constructive institutions---and develops its metatheory from that requirement. The kernel theory is a dependent type theory with cumulative universes, an impredicative universe of propositions, inductive families, and an intensional identity type. Proof irrelevance is an axiom with an explicit witness rather than a conversion rule, so proofs of a proposition are provably equal yet remain distinct terms. Definitional equality is joinability under an orthogonal rewriting system---confluent, injective on type formers, and decidable by a type-free test---while a separate subsumption judgment covers universe cumulativity, codomain-covariant function types, and declared class inclusions. We prove consistency via a set-theoretic model; develop a justification logic over the kernel's propositions to track evidence and support; organize environments as an institution so that store extensions are signature morphisms and external verdicts enter via comorphisms; formulate a typed knowledge-graph theory with records, classes, and refinements; and prove the correctness of conversion, normalization by evaluation, and bidirectional type checking relative to termination. A Lean 4 formalization accompanies the volume.
#05Nov 5, 2026
cs.LG
Permanent and Transient Representations for Continual Reinforcement Learning
Nishanth Anand, Doina Precup
Continual Reinforcement Learning (CRL) agents struggle to adapt to new situations while retaining prior knowledge, leading to a stability-plasticity trade-off. One approach to solving this problem is by using complementary learning systems: one for slow, long-term learning and another for quick, transient adaptation. Anand & Precup (2023) instantiated this idea using permanent and transient value functions, but used the same features in both systems. Intuitively, long-term learning should rely more on parametric representations for broad generalization, while transient learning would be best served by non-parametric representations for situation-specific adaptation. In this paper, we explore this idea both theoretically and empirically. By introducing a novel non-parametric approximation for estimating the transient value function in complex tasks, we demonstrate that our method enables online learning and outperforms competitive baselines on image-based tasks and Craftax-Classic, underscoring the effectiveness of our system-level decomposition.