A.1 Neural Hybrid Automata: Modules and Hyperparameters We provide a notation and summary table for Neural Hybrid Automata (NHA). The table serves as a quick reference for the core concepts introduced in the main text. Labels every subjtrajectory Xi with a mode z to ensure mode-conditioned decoder Fz can reconstruct it despite Neural ODE representation limitations (uniqueness of solutions given an initial condition). The only NHA hyperparameter beyond module architectural choices is m, or number of latent modes provided to the model at initialization. Performance effects of changing mhave been explored in Section 5.2 and Appendix B.2. Appendix B.2 further provides analyzes potential techniques to prune additional modes. A.2 Gradient Pathologies We provide some theoretical insights on the phenomenon of gradient pathologies with the simple example of a one-dimensional linear hybrid system with two modes and one timed jump, xt = axtt<τ bxtt>= τ t 6= τ x+t = cxtt= τ (A.1)

A.1 MCTS-kSubS algorithm In Algorithm 4 we present a general MCTS solver based on AlphaZero. Solver repeatedly queries the planner for a list of actions and executes them one by one. Baseline planner returns only a single action at a time, whereas MCTS-kSubS gives around kactions - to reach the desired subgoal (number of actions depends on a subgoal distance, which not always equals k in practice). MCTS-kSubS operates on a high-level subgoal graph: nodes are subgoals proposed by the generator (see Algorithm 3) and edges - lists of actions informing how to move from one subgoal to another (computed by the low-level conditional policy in Algorithm 2). The graph structure is represented by treevariable. For every subgoal, it keeps up to C3 best nearby subgoals (according to generator scores) along with a mentioned list of actions and sum of rewards to obtain while moving from the parent to the child subgoal. Most of MCTS implementation is shared between MCTS-kSubS and AlphaZero baseline, as we can treat the behavioral-cloning policy as a subgoal generator with k = 1. MCTS-kSubS and the baseline are encapsulated in GEN_CHILDREN function (Algorithms 5 and 6).

With the popularity of automatic code generation tools, such as Copilot, the study of the potential hazards of these tools is gaining importance. In this work, we explore the social bias problem in pre-trained code generation models. We propose a new paradigm to construct code prompts and successfully uncover social biases in code generation models. To quantify the severity of social biases in generated code, we develop a dataset along with three metrics to evaluate the overall social bias and fine-grained unfairness across different demographics. Experimental results on three pre-trained code generation models (Codex, InCoder, and CodeGen) with varying sizes, reveal severe social biases. Moreover, we conduct analysis to provide useful insights for further choice of code generation models with low social bias1.

Sparse identification of nonlinear dynamics (SINDy) has been widely used to discover the governing equations of a dynamical system from data. It uses sparse regression techniques to identify parsimonious models of unknown systems from a library of candidate functions. Therefore, it relies on the assumption that the dynamics are sparsely represented in the coordinate system used. To address this limitation, one seeks a coordinate transformation that provides reduced coordinates capable of reconstructing the original system. Recently, SINDy autoencoders have extended this idea by combining sparse model discovery with autoencoder architectures to learn simplified latent coordinates together with parsimonious governing equations. A central challenge in this framework is robustness to measurement error. Inspired by noise-separating neural network structures, we incorporate a noise-separation module into the SINDy autoencoder architecture, thereby improving robustness and enabling more reliable identification of noisy dynamical systems. Numerical experiments on the Lorenz system show that the proposed method recovers interpretable latent dynamics and accurately estimates the measurement noise from noisy observations.

The original CLUTRR data generation framework made sure that each testproof is not in the training set in order to test whether a model is able to generalize to unseen proofs. Initial results on the original CLUTRR test sets resulted in strong model performance ( 99%) on levels seen during training (2, 4, 6) but no generalization at all ( 0%) to other levels. The models are given as input " [story] [query] " and asked to generate the proof and answer. Models are trained on levels2,4,6only. In our case, the entity names are important to evaluate systematic generalization.