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In this paper, we revisit the investigation of solitary-wave interactions in the nonlinear Schrödinger model, both in the presence and absence of a parabolic trapping potential. While approximate dynamics, based on variational—or similar—methods, governed by a system of ordinary differential equations (ODEs) for both bright and dark-soliton interactions have been well established in the literature based on physical expert considerations, this study focuses on a data-driven approach, the so-called sparse identification of nonlinear dynamics. Accordingly, our purpose is to use partial differential equation (PDE) time-series of select waveform diagnostics in order to numerically reconstruct such approximate dynamics, without prior knowledge thereof. The purpose is not only to verify the robustness of the dynamical approximated ODEs, but also to shed light on the application of such a data-driven methodology in the study of soliton interactions and to formulate a complementary approach, more reliant on the wealth of PDE data and less so on expert theoretical constructs. 

We introduce cumulative tic-tac-toe, a novel variant of the classic 3×3 tic-tac-toe game in which play continues until the board is completely filled. Each player’s final score is determined by the total number of three-in-a-row sequences they form. Using combinatorial game theory (CGT), we establish that under optimal play, the game is a draw, and we characterize its theoretical properties. To empirically validate and optimize practical play, we develop a reinforcement learning (RL) framework based on temporal-difference (TD) learning, which is enhanced with a domain-informed evaluation function to accelerate convergence. The experimental results show that our triplet-coverage difference (TCD) evaluation function reduces the average number of training episodes by approximately 23.1% compared with a random-initialization baseline, a statistically significant improvement at the 5% significance level. These results demonstrate the efficiency of our CGT–RL approach for cumulative tic-tac-toe and suggest that similar methods may be useful for analyzing related combinatorial games. We also discuss potential analogies in domains such as competitive resource allocation and coalition formation, illustrating how cumulative-scoring games connect abstract game-theoretic ideas to practical sequential decision problems. 

Synthetic datasets are increasingly used in education research for methodological validation, privacy-preserving data sharing, and reproducible equity analysis; however, most generative approaches prioritize marginal distributional similarity without ensuring preservation of multilevel inferential properties. This limitation is consequential for repeated-measures data analyzed using intersectionality-focused hierarchical models, where conclusions depend on variance partitioning, partial pooling, and stratum-level heterogeneity. We introduce MAI-GAN, a hybrid generative framework that implements a structure–residual decomposition approach combining Bayesian longitudinal MAIHDA with conditional GAN-based residual generation. Inferential fidelity is operationalized with respect to multilevel intersectional models by explicitly targeting the preservation of fixed effects, variance components, and variance partitioning coefficients, while baseline composition is maintained via stratified bootstrap resampling. Applied to a six-semester undergraduate biology dataset (N = 2669 students), MAI-GAN was evaluated across multiple independent random seeds and consistently reproduced baseline-dependent residual structure and key inferential quantities. These results demonstrate that model-aligned generative strategies can produce synthetic longitudinal datasets that remain coherent under intersectionality-focused multilevel analysis, offering a principled foundation for equity-oriented synthetic data generation. 

Abstract. We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon–Heiles system, i.e., a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon–Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg–de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded [Formula: see text] regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon–Heiles system and the Korteweg–de Vries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems. 

We propose a data-driven framework for learning reduced-order moment dynamics from partial differential equation (PDE)-governed systems using neural Ordinary Differential Equations (neural ODEs). In contrast to derivative-based methods, such as SINDy (Sparse Identification of Nonlinear Dynamics), which necessitate densely sampled data and are sensitive to noise, our approach based on neural ODEs directly models moment trajectories, enabling robust learning from sparse and potentially irregular time series. Using as an application platform the nonlinear Schrödinger equation, the framework accurately recovers governing moment dynamics when closure is available, even from limited, irregular, and noisy observations. For systems without analytical closure, we introduce a data-driven coordinate transformation strategy based on Stiefel manifold optimization, enabling the discovery of low-dimensional representations in which the moment dynamics become closed, facilitating interpretable and reliable modeling. We also explore cases where a closure model is not known, such as a Fisher–Kolmogorov–Petrovskii–Piskounov reaction–diffusion system. Here, we demonstrate that neural ODEs can still effectively approximate the unclosed moment dynamics and achieve superior extrapolation accuracy compared to physical-expert-derived ODE models. This advantage remains robust even under sparse and irregular sampling, highlighting the method’s robustness in data-limited settings. Our results highlight the neural ODE framework as a powerful and flexible tool for learning interpretable, low-dimensional moment dynamics in complex PDE-governed systems. 

Abstract This paper demonstrates the robustness of Lipschitz-regularized $$\alpha $$-divergences as objective functionals in generative modeling, showing they enable stable learning across a wide range of target distributions with minimal assumptions. We establish that these divergences remain finite under a mild condition—that the source distribution has a finite first moment—regardless of the properties of the target distribution, making them adaptable to the structure of target distributions. Furthermore, we prove the existence and finiteness of their variational derivatives, which are essential for stable training of generative models such as generative adversarial networks and gradient flows. For heavy-tailed targets, we derive necessary and sufficient conditions that connect data dimension, $$\alpha $$ and tail behaviour to divergence finiteness, that also provide insights into the selection of suitable $$\alpha $$’s. We also provide the first sample complexity bounds for empirical estimations of these divergences on unbounded domains. As a byproduct, we obtain the first sample complexity bounds for empirical estimations of these divergences and the Wasserstein-1 metric with group symmetry on unbounded domains. Numerical experiments confirm that generative models leveraging Lipschitz-regularized $$\alpha $$-divergences can stably learn distributions in various challenging scenarios, including those with heavy tails or complex, low-dimensional, or fractal support, all without any prior knowledge of the structure of target distributions. 

Despite the remarkable fidelity of eukaryotic DNA replication, nucleotide misincorporation errors occur in every replication cycle, generating mutations that drive genetic diseases and genome evolution Here, we show that transcription factor (TF) proteins, key players in gene regulation, can increase mutagenesis from replication errors by directly competing with the recognition of DNA mismatches by MutS_alpha, the primary initiator of eukaryotic mismatch repair (MMR) , We demonstrate this TF-induced mutagenesis mechanism using a yeast genetic assay that quantifies the accumulation of mutations in TF binding sites Analysis of human cancer mutations recaptitulate the trends observed in yeast, with mutations arising from MYC-bound mismatches being enriched in MMR-proficient cells. These findings implicate TF-MMR competition as a critical determinant of somatic hypermutation at TF binding sites in cancer. Furthermore, our results provide a molecular mechanism for the higher-than-expected rate of rate genetic variants at TF binding sites, with important implications for regulatory DNA evolution.