scientific programming

This paper identifies a critical implicit bias arising from numerical integration scheme selection in learning dynamical systems from sparse temporal observations: even with perfect data fit, inappropriate schemes (e.g., explicit Euler) can fundamentally mischaracterize system dynamics—such as misidentifying a true damped oscillator as exhibiting “anti-damping” and reverse oscillation. Method: We formulate a unified optimization framework to systematically analyze modeling bias induced by diverse numerical integrators (explicit/implicit, low-/high-order) in system identification. Contribution/Results: Through rigorous theoretical analysis and empirical validation, we demonstrate that such numerical artifacts can yield physically contradictory conclusions. Crucially, this work is the first to explicitly identify, formalize, and quantify the threat posed by numerical pseudospectra to the reliability of dynamical system learning—thereby providing both a foundational warning and methodological grounding for trustworthy physics-informed machine learning.

Subgrid-scale errors in chaotic dynamical systems modeling induce long-term prediction instability and statistical distortions. Method: This paper proposes an explicit bidirectional (fine-to-coarse and coarse-to-fine) multiscale data-driven modeling framework that integrates dynamical system embedding, chaotic time-series modeling, and physics-constrained learning. Contribution/Results: It introduces the first neural architecture explicitly designed to preserve both multiscale information flow and numerical stability. Applied to climate subgrid-scale parameterization, the method enables physically consistent error correction. Experiments demonstrate substantial improvements in long-term forecast stability and statistical fidelity—quantified via invariant measures and temporal correlations—while effectively mitigating the “missing physics” problem arising from unresolved small-scale processes in baseline chaotic models.

This study addresses the challenge of long-term, high-precision forecasting of low-dimensional chaotic systems under noise-free observations. We propose an ordinary least squares (OLS) modeling approach leveraging high-degree polynomial features and 512-bit arbitrary-precision arithmetic—deliberately avoiding black-box models such as neural networks. By rigorously controlling both numerical error and model bias, our method achieves machine-precision trajectory prediction (≈10⁻⁶⁴) for canonical systems including the Lorenz-63, Thomas, and Lorenz-96 attractors—the first such result to our knowledge. Effective prediction horizons extend to 32–105 Lyapunov times, substantially exceeding the theoretical limit of ~13 Lyapunov times attained by conventional methods. These findings demonstrate that, for suitably simplified real-world chaotic systems, dynamics can be learned and reconstructed nearly analytically, providing critical empirical evidence for the solvability boundary of chaotic modeling.

This work addresses the challenge of predicting chaotic dynamics across multiple scenarios in the CTF-4-Science Lorenz benchmark—including clean prediction, noisy reconstruction, prediction from noisy inputs, few-shot learning, and parameter generalization—by proposing a scenario-specific modeling strategy that abandons the pursuit of a single universal model. The approach integrates specialized modules tailored to each task: trajectory smoothing for denoising, NG-RC/NVAR-based attractor prediction, sensitive-prefix-guided Lorenz dynamical correction, and parametric prefix interpolation. Each component is optimized according to the characteristics of its target scenario. Evaluated on the benchmark, this method achieves a public score of 79.63, substantially outperforming generic models and demonstrating the efficacy and superiority of a scenario-aware, specialized modeling paradigm for chaotic system prediction.

Doctoral students in life sciences commonly lack formal software engineering training, hindering the development of robust, reproducible, and collaborative research software. Method: This study proposes ten pedagogical principles for research software development, establishing the first systematic framework centered on “research software pedagogy”—distinct from generic programming instruction. It integrates software engineering best practices (e.g., Git-based version control, CI/CD pipelines, unit testing, RESTful API design), learning science principles, and authentic research workflows, emphasizing the seamless embedding of automation, documentation, testing, and collaborative practices throughout the research lifecycle. Contribution/Results: The framework delivers a generalizable, plug-and-play pedagogical paradigm. Deployed across multiple Chinese universities’ life sciences PhD programs, it has demonstrably improved software deliverable quality, code reusability, and cross-team collaboration efficiency—bridging critical gaps between computational literacy and rigorous, team-based scientific software practice.

This work investigates whether pretrained image editing models can serve as a universal interface for solving diverse physical equations. The approach encodes both inputs and solutions of physical problems as images, incorporates lightweight adapters to embed scalar parameters, and trains the model under a unified architecture using numerical or analytical solutions across multiple equation types—including elliptic, heat, and Navier-Stokes equations. For the first time, it systematically demonstrates that general-purpose generative models can effectively represent both static and dynamic physical mappings, even capturing shocks and unstable phenomena, thereby expanding their applicability in scientific computing. Experiments across more than ten problem classes yield promising results, yet also reveal limitations of image-based representations in handling wide numerical ranges, enforcing constraints, and simulating long-term chaotic dynamics, such as those in the Kuramoto–Sivashinsky equation.

Scientific machine learning experiments often suffer from distorted performance evaluations due to poor experimental design and inconsistent documentation. To address this, we propose a principled framework for ML experimentation tailored to scientific research, encompassing data preprocessing, model selection, cross-validation, and reporting—emphasizing reproducibility, fair comparison, and transparency. Our key contributions include two novel quantitative metrics: the Logarithmic Overfitting Ratio (LOR) and Composite Overfitting Score (COS), which jointly characterize overfitting severity and instability across cross-validation folds. Complementing these, we introduce standardized preprocessing protocols, rigorously defined strong baselines, and modular visualization templates for diagnostic analysis. Empirical evaluation demonstrates that our framework substantially enhances experimental rigor, reproducibility, and result credibility in scientific ML. It further enables robust performance assessment and cross-study comparability, providing systematic support for establishing reliable benchmarks.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.

This work addresses the limitations of current automatic formalization research, which predominantly focuses on well-supported mathematical domains and relies solely on kernel acceptance rate as a quality metric, thereby neglecting the practical needs of underrepresented areas such as numerical analysis and lacking comprehensive evaluation. For the first time, we employ a Lean 4 coding agent to formalize an entire textbook—*Numerical Methods for Ordinary Differential Equations*—from scratch and introduce a three-dimensional evaluation framework that jointly assesses semantic correctness, Mathlib reusability, and cross-file reusability. Through LLM-as-judge, semantic validation, and dependency analysis, we uncover pervasive issues in existing systems, including incomplete statements and weakened assumptions, demonstrating that kernel acceptance rate substantially overestimates formalization quality. Our approach establishes a reproducible, multidimensional auditing paradigm for trustworthy automated formalization.

This work addresses the lack of intuitive, immediate feedback on fitting errors in existing model-fitting approaches. It proposes an interactive fitting framework that integrates visual and auditory feedback: as users manipulate parametric curves, the system synthesizes audio in real time, with greater model-data discrepancies producing louder and more dissonant sounds. This is the first approach to incorporate auditory cues into model exploration, enabling multisensory assessment of fit quality. Combining interactive visualization, real-time audio synthesis, and Gaussian process regression, the method demonstrates effectiveness and generalizability across four diverse case studies—golf putting, dilution experiments, cosmological parameter estimation, and temperature data fitting—significantly enhancing users’ intuitive perception of model misfit.