Reinforcement Learning with Verifiable Rewards has shown great promise in enhancing mathematical reasoning in large language models. However, existing approaches provide limited intermediate supervision for complex, multi-step reasoning tasks. To bridge this gap, we propose \textbf{Hierarchical Hint-based Reward}, a framework that leverages the intrinsic sub-question structure of mathematical problems as structured intermediate conditioning signals during RLVR training. Unlike chain-of-thought prompting, these hierarchical hints serve as logically grounded intermediate milestones that effectively reduce the reasoning horizon toward the final answer. To validate this framework, we introduce \textbf{HiMath}, a benchmark of hierarchically structured mathematical problems. Across diverse LLMs, hierarchical hints consistently improve reasoning performance even without task-specific training. When integrated into RL training via H$^2$R, they boost Qwen2.5-Math-7B accuracy on HiMath from 30.4\% to 39.1\% using hierarchical hints, and up to 45.6\% with Oracle Hierarchical Hints. This pronounced improvement demonstrates that incorporating reliable intermediate checkpoints into reward design is essential for sustaining complex mathematical reasoning.
We show that established performance metrics in binary classification, such as Matthews' correlation coefficient (MCC), Cohen's $κ$, the F-score or the Jaccard similarity coefficient are not robust to class imbalance in the sense that if the proportion of the minority class tends to $0$, the true positive rate (TPR) of the Bayes classifier under these metrics {becomes very small or even tends to $0$}. Thus, in imbalanced classification problems, these metrics favour classifiers which ignore the minority class. To alleviate this issue we introduce robustified modifications of the MCC, of Cohen's $κ$ and of the F-score with an additional tuning parameter which allows to adapt the amount of robustness against class imbalance. As theoretical guarantee we show that the Bayes-optimal classifier for these robustified performance metrics, when expressed in terms of the density ratio $f_1/f_0$ of the class-conditional densities $f_i$, has a threshold parameter which is upper-bounded in terms of the tuning parameters. Therefore, even in strongly imbalanced settings, the TPR associated to this classifier will be bounded away from $0${, provided that $f_1/f_0$ exceeds this bound with positive probability under the positive class}. We numerically illustrate the behaviour of the various performance metrics and the effect of the tuning parameters in simulations as well as on a credit default data set. We also discuss connections to the receiver operating characteristic and precision-recall curves, which provide an alternative perspective on the proposed notion of robustness, and give recommendations on how to combine their usage with performance metrics.
Nonlinear structural system identification remains challenging when governing physics is only partially known and measurements are sparse. Deep Bayesian filtering addresses these challenges by combining learnable dynamical representations with probabilistic state-space models, physical inductive biases, and uncertainty-aware inference. This paper presents, for the first time, a unified tutorial treatment of deep Bayesian filtering for nonlinear structural system identification, connecting model formulation, learning, posterior inference, and evaluation. Amortized variational inference and differentiable Kalman filtering are introduced as complementary strategies within this broader framework. The Deep Markov Model (DMM) provides a data-driven reference, while the Physics-guided Deep Markov Model (PgDMM) and Neural Extended Kalman Filter (Neural EKF) serve as illustrative instances through which methodological choices and their implications are examined. Dynamic state estimation and joint state and parameter estimation are demonstrated on a three-dimensional shear-frame benchmark with hysteretic Bouc-Wen connections and sparse acceleration measurements. The comparative analysis yields new observations concerning the relationships among reconstruction accuracy, predictive fidelity, physical interpretability, and estimation robustness. Physical structure improves latent-state interpretability without necessarily ensuring faithful predictive dynamics, while comparable typical reconstruction accuracy can conceal substantial differences in estimation robustness. These findings motivate practical guidelines for model construction, training, and evaluation that distinguish reconstruction, forecasting, state recovery, and parameter identification. The tutorial concludes by outlining challenges in uncertainty calibration, identifiability, and validation on real structures.
We measure the emotion geometry of six LLMs spanning Google's Gemma 2 family. Using an extraction pipeline established by Sofroniew et al. (2026) on Claude Sonnet 4.5 and replicated by Hollowell (2026) on Gemma 2 27B base, we compare emotion geometry across base and instruction-tuned variants of Gemma 2 2B, 9B, and 27B. First, all six model cells exhibit the core emotion geometry. A linear map fit between two cells places held-out emotions nearest their own vectors in the target model. Across all fifteen model pairs, fit in one direction each, this failed exactly once, and the median rank was 1 in every pair. Linear maps refit on cleaned embedding-table vectors recover every held-out emotion at rank 1 in all thirty ordered pairs. Second, scale spreads the geometry's variance across more directions without weakening its tie to valence. The leading component's share of variance falls from 31.6 percent (2B) to 29.8 (9B) to 26.7 (27B) in the base models and from 31.9 (2B) to 29.2 (9B) to 27.3 (27B) in the tuned ones, while its correlation with human valence ratings stays between .71 and .78 in all six models. Third, we project each emotion vector onto the output vocabulary and ask at what depth the vector's own emotion word enters the top 10,000 of the 256,000 tokens. In the 2B and 9B models the median emotion word is there from layer 0, while in the 27B models it begins near chance, stays outside the top 10,000 for half the network, and enters at layer 23 in one step. Fourth, tuning leaves every registered analysis-layer measure within its floor. At every size, valence correlation rises and arousal correlation falls by amounts within their floors, the latter against prediction. PC2's variance share also falls, with a pooled decrease significant after correction. Mid-late layer stability exceeds its floor and traces to base 27B's final layer. Code, data, and audit reports are released.
A spatially aware dictionary-free eigenfunction discovery (SADFED) framework is proposed for identification of low-rank Koopman models from data without prescribing a lifting dictionary, kernel, or neural-network eigenfunction architecture. A reference trajectory is selected and used to determine the Koopman modes by regularized least squares (LS). Then, a transformed temporal basis allows the eigenfunction values at all sampled initial conditions to be obtained by a second regularized LS projection. Consequently, only the real and imaginary parts of the eigenvalues remain as the optimization variables. Interpolation of the identified eigenfunction samples reveals their spatial structure, enabling numerical estimation of their gradients. A joint objective combines trajectory reconstruction error with a normalized Koopman partial differential equation (KPDE) residual, promoting spatial consistency with the KPDE over the sampled region and serving as a physics-informed regularizer. The method is evaluated on a system with analytical Koopman eigenfunctions, the FitzHugh-Nagumo system, the van der Pol oscillator, the Duffing system, and a two-spool turbojet engine. The examples demonstrate recovery of known eigenfunctions, sensitivity to reference trajectory and hyperparameters, limit-cycle harmonics and isochrons, discontinuous indicator eigenfunctions and isostables, symmetry exploitation, and construction of state-dependent lifted input dynamics. For the turbojet example, the identified model is further used for state estimation and design of a gain-scheduled tracking linear quadratic Gaussian controller. The results indicate the applicability of SADFED to Koopman spectral identification and control-oriented modeling of nonlinear dynamical systems.