Kenneth ChengMathematical modeling

Frontier model studio

Use a small model to expose a large research question.

Six browser experiments translate active research ideas into mechanisms that can be changed, challenged, and extended. They are conceptual laboratories, not claims that the open problem has been solved.

Choose an experimentBrowse all model labs

Experiment map

Start from the question.

Each experiment isolates one mechanism. The labels identify the mathematical lens; the question tells you what to change first.

  1. 01Critical transitionsWhen does gradual pressure trigger an abrupt change?Bifurcation theoryIntermediate
  2. 02Equation discoveryCan a governing equation be recovered from noisy observations?Sparse regressionAdvanced
  3. 03Collective motionHow can local alignment create global order?Agent-based systemsIntermediate
  4. 04Higher-order contagionWhat changes when groups, not only pairs, transmit influence?Simplicial dynamicsAdvanced
  5. 05Operator learningWhat must a model learn to map one function into another?Scientific machine learningAdvanced
  6. 06Persistent homologyWhich shapes survive a change of observational scale?Topological data analysisAdvanced

01 · Bifurcation theory

Critical transitions and path dependence

x′ = a + x − x³

A single control parameter tilts a potential landscape. Stable states can disappear at a fold, so a slowly changing pressure can produce a sudden jump and a different return path.

Resulting state
—
Stable wells
—
Local recovery rate
—

Research promptsMove a slowly from negative to positive, then reverse it. Does the state retrace the same path?Near which pressure does recovery become slow enough to resemble an early-warning signal?

This normal form demonstrates a mechanism, not a forecast. Empirical early-warning indicators can fail when observations do not satisfy the assumed bifurcation structure.

Research basis [1] [2] [3]

02 · Data-driven dynamics

Discovering an equation from noisy data

x′ ≈ Θ(x)ξ

Sparse identification starts with a library of candidate terms, then asks for the smallest equation that explains the measured derivative. Here the hidden law is x′ = x − x³.

Terms retained
—
Curve RMSE
—
Recovered law
—

Research promptsCan more observations compensate for a poor sparsity threshold?When does a visually accurate curve still produce the wrong symbolic equation?

The demonstration uses direct derivative observations and a polynomial library. Real experiments must also handle differentiation error, hidden variables, correlated terms, and model-library misspecification.

Research basis [4] [5]

03 · Active matter

Collective motion from local alignment

θᵢ(t + 1) = arg Σⱼ∈Nᵢ eⁱθʲ + ηξᵢ

Self-propelled particles see only nearby headings. Repeated local alignment can break rotational symmetry and produce coherent global motion even without a leader.

Global order Φ
—
Mean neighbours
—
Observed regime
—

Research promptsDoes increasing the interaction radius always compensate for noise?How would obstacles, mixed species, or delayed responses change the transition?

This is a seeded, finite Vicsek-style simulation. Its apparent transition depends on density, boundary conditions, system size, and the chosen noise rule.

Research basis [6] [7]

04 · Higher-order networks

Contagion through pairs and groups

ρ′ = −ρ + λ₁ρ(1 − ρ) + λ₂ρ²(1 − ρ)

Pairwise contact contributes a term proportional to ρ; group reinforcement contributes a nonlinear ρ² term. That extra order can create critical-mass effects and abrupt adoption.

Selected final state
—
Small-seed final state
—
Seed sensitivity
—

Research promptsFind a parameter set where 2% adoption collapses but a larger seed persists.Which empirical observations would distinguish pairwise influence from genuine group reinforcement?

This mean-field social-contagion model is not an infectious-disease forecast. Group interaction, memory, network structure, and causal identification require separate evidence.

Research basis [8] [9]

05 · Scientific machine learning

Learning maps between functions

𝒢: u₀(x) ↦ u(x,t)

Operator learning asks for one reusable map across a family of inputs, rather than one solution vector on one grid. This experiment isolates the spectral representation problem using the heat operator.

Relative L² error
—
Spectrum retained
—
Input family
—

Research promptsWhy do rough inputs demand more modes at short prediction times?What additional tests are needed before claiming resolution invariance or out-of-distribution generalisation?

No neural network is trained here. The exact heat operator and a truncated Fourier representation expose the approximation question that architectures such as FNOs attempt to learn from data.

Research basis [10] [11]

06 · Topological data analysis

Finding shape across scales

β₀ = components · β₁ = loops

A Vietoris–Rips filtration connects points within a distance ε and fills every completed triangle. Features that survive across many scales are candidates for signal rather than sampling noise.

Components β₀
—
Loops β₁
—
Edges / triangles
—

Research promptsAt which scales do the intended loops exist, and how long do they persist?How should statistical uncertainty enter a claim that one topological feature is “real”?

The displayed Betti numbers are computed for the current finite Rips complex. A research analysis would examine the full persistence diagram, sampling variability, and a domain-specific null model.

Research basis [12] [13]

References

From mechanism to literature.

Foundational formulations are paired with recent extensions or empirical limitations. The references support the research framing; they do not validate every simplification used in the browser model.

  1. Scheffer, M., et al. (2009). Early-warning signals for critical transitions. Nature, 461, 53–59. https://doi.org/10.1038/nature08227
  2. Dylewsky, D., Anand, M., & Bauch, C. T. (2024). Early warning signals for bifurcations embedded in high dimensions. Scientific Reports, 14, 18277. https://doi.org/10.1038/s41598-024-68177-1
  3. O’Brien, D. A., et al. (2023). Early warning signals have limited applicability to empirical lake data. Nature Communications, 14, 7942. https://doi.org/10.1038/s41467-023-43744-8
  4. Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932–3937.
  5. Rosafalco, L., Conti, P., Manzoni, A., Mariani, S., & Frangi, A. (2024). EKF–SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 431, 117264.
  6. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75(6), 1226–1229.
  7. Lardet, E., Chen, L., & Bertrand, T. (2026). Flocking beyond one species: Novel phase coexistence in a generalized two-species Vicsek model. Physical Review Letters, accepted paper. https://doi.org/10.1103/lt6f-yjpd
  8. Iacopini, I., Petri, G., Barrat, A., & Latora, V. (2019). Simplicial models of social contagion. Nature Communications, 10, 2485.
  9. Lin, Z., et al. (2024). Higher-order non-Markovian social contagions in simplicial complexes. Communications Physics, 7, 175.
  10. Li, Z., et al. (2021). Fourier neural operator for parametric partial differential equations. International Conference on Learning Representations.
  11. Cao, Q., Goswami, S., & Karniadakis, G. E. (2024). Laplace neural operator for solving differential equations. Nature Machine Intelligence, 6, 631–640.
  12. Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2002). Topological persistence and simplification. Discrete & Computational Geometry, 28(4), 511–533.
  13. Chen, Y., et al. (2024). EMP: Effective multidimensional persistence for graph representation learning. Proceedings of the Second Learning on Graphs Conference, PMLR 231, 24:1–24:12.