A Scientometric Review of Neural Differential Equations: Mapping Research Impact and Collaborative Networks
Bibliographic record
Abstract
Neural Differential Equations (NDEs) represent an emerging class of machine learning models that combine the strengths of neural networks and differential equations to model continuous-time dynamics with high interpretability. This study presents a comprehensive bibliometric analysis of NDE research using data extracted from the Scopus database. Analytical tools such as Biblioshiny, VOSviewer, and CiteSpace were employed to uncover patterns, trends, and structural relationships within the field. The annual scientific production shows a significant growth trajectory, with a peak in 2023, indicating rising scholarly interest. Most relevant authors include Rackauckas, Nopens, and Chien, whose contributions have shaped the theoretical and applied dimensions of NDEs. Co-citation networks of both authors and journals revealed well-defined research clusters focused on deep learning techniques, scientific machine learning, and graph-based modeling. Country-wise analysis highlights the dominance of the United States and China, followed by notable contributions from the UK, Canada, and India. Keyword co-occurrence and trend analysis identified emerging themes such as “transformer,” “graph neural networks,” and “scientific machine learning,” reflecting ongoing methodological innovation. Thematic evolution and mapping show a shift from foundational terms to more specialized and interdisciplinary applications. Identified research gaps suggest a need for stronger theoretical integration, benchmark development, and application in real-world domains, offering practical implications for researchers and practitioners in AI, engineering, and applied sciences.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.021 | 0.117 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.106 | 0.160 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.006 | 0.009 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".