Neural Symbolic Regression Boosts Network Dynamics Prediction Accuracy by Up to 60%

Neural Discovery of Network Dynamics (ND²)

The researchers from Tsinghua University propose a neural‑symbolic regression method called ND² (Neural Discovery of Network Dynamics) that automatically discovers mathematical formulas describing system dynamics from data. By defining a set of network dynamical operators—source operator φ(s), target operator φ(t), and aggregation operator ρ—the original high‑dimensional symbolic search is reduced to an equivalent one‑dimensional problem, eliminating the exponential growth of the search space with network size.

NDformer‑Guided Symbolic Search

The method incorporates an NDformer‑guided symbolic search algorithm. The architecture consists of a neural module (NDformer) that learns latent features of system dynamics and predicts probability distributions over symbols, and a symbolic module (Monte‑Carlo Tree Search, MCTS) that selects symbols according to these probabilities to construct candidate formulas. For each candidate, a reward calculator fits any unknown coefficients with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm and returns a score that balances accuracy and simplicity, steering MCTS toward shorter, more accurate expressions.

NDformer itself fuses a Graph Neural Network (GNN) with a Transformer, pre‑trained to predict the next symbol of a formula given the network structure and node activity data. This pre‑training enables the model to capture complex dynamical features and to guide MCTS efficiently through the reduced search space.

Experimental Validation Across Scales

To verify ND², the team applied it to diverse complex systems ranging from cellular to urban scales. In a yeast cell‑division gene‑expression network, the discovered formula improved prediction accuracy by roughly 60% compared with existing empirical formulas and revealed higher‑order interactions where a third gene modulates the regulation between two genes.

In a microbial ecological network, the ND²‑derived dynamics outperformed the traditional Lotka‑Volterra model by about 56% and exhibited a novel behavior: populations with larger abundances are less affected by other species, a pattern absent in classic models.

For epidemic spreading, ND² was used on seven representative regions spanning city‑level to global‑scale mobility networks. The method automatically uncovered high‑precision transmission equations and highlighted distinct mechanisms: in the United States, the self‑evolutionary dynamics remain stable, whereas in China the transmission intensity weakens as case numbers rise, reflecting effective containment policies. Moreover, inter‑regional coupling is strong across U.S. states but weak among Chinese provinces, matching the differing travel restrictions.

Macro‑Level Steady‑State Analysis

Using the discovered equations, the researchers analyzed macroscopic steady‑state properties. In China, infection numbers remain controllable when inter‑provincial traffic stays below a threshold, but surge dramatically once the threshold is crossed, indicating a critical transition. In the United States, average infections increase linearly with cross‑state traffic, suggesting a smoother impact of mobility controls.

Broader Implications

Through cross‑scale, multidisciplinary validation, the study demonstrates that neural symbolic regression not only reliably discovers accurate dynamical formulas but also uncovers hidden microscopic mechanisms in complex systems, offering a powerful new tool for fundamental scientific discovery.

Reference: "Discovering network dynamics with neural symbolic regression", Nature Computational Science (2025).