Kenneth ChengMathematical modeling

Research inquiry studio

Move from an interesting pattern to a defensible research question.

These laboratories focus on what a model can identify, what an experiment can estimate, and what a decision can withstand. Each interaction is paired with assumptions, claim limits, a complete article, and traceable academic references.

Choose an inquiryReturn to frontier models

Inquiry map

Start from the inferential difficulty.

The three questions occur after a model has been written down. They ask whether its parameters, causal effects, or decisions are actually supported by the available information.

  1. 01IdentifiabilityCan the data distinguish the parameters that produced them?Inverse problemsResearch design
  2. 02Network interferenceWhat is a treatment effect when one unit changes another?Causal inferenceResearch design
  3. 03Robust decisionsWhich decision survives a plausible change in the distribution?Distributionally robust optimizationResearch design

A research-ready loop

Do not stop when the graphic looks convincing.

A useful interaction should reveal the next experiment or assumption that could overturn the result.

  1. 01Declare the estimand

    State the parameter, causal contrast, or decision criterion before changing the controls.

  2. 02Stress the information

    Reduce measurement range, add spillovers, or enlarge the ambiguity set.

  3. 03Find the failure boundary

    Record when an apparently stable conclusion becomes non-unique, biased, or too conservative.

  4. 04Design the next evidence

    Choose a new measurement, exposure mapping, or validation distribution that can discriminate competing explanations.

01 · Inverse problems

Good fit, uncertain parameters

y(S) = VmaxS/(Km + S)

A Michaelis–Menten response can be fitted closely by many parameter pairs when observations cover only a narrow substrate range. The heatmap shows the resulting loss surface rather than hiding it behind one best-fit estimate.

Best Vmax
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Best Km
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Near-optimal grid
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Research promptsHow far must the substrate range extend before the low-loss ridge contracts around the true pair?Would an extra replicate or a measurement at a new substrate level reduce uncertainty more efficiently?

This is a deterministic synthetic demonstration of practical identifiability. It does not perform profile-likelihood inference or prove structural identifiability for a general dynamical system.

Research basis [1] [2] [3]

Read the full article: When the Fit Is Not the Model →

02 · Causal inference

Treatment effects with spillovers

Yi = αi + τZi + γEi

The exposure Ei is the treated fraction of unit i's neighbours. Once γ is nonzero, an untreated unit can change because its neighbours were treated, and a difference in means no longer isolates the direct effect τ.

Naive difference
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Direct effect τ
2.00
All-treated policy effect
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Research promptsCan two assignment patterns with the same treatment coverage produce different naive estimates?Which exposure mapping would match a classroom, household, marketplace, or social-network intervention?

The graph, outcomes, and exposure rule are synthetic and known. Real studies must justify the interference neighbourhood, treatment design, positivity, and the estimand before choosing an estimator.

Research basis [4] [5] [6]

Read the full article: When Treatment Spills Over →

03 · Optimization under uncertainty

A decision that survives probability shift

minq maxp ∈ P(ρ) Ep[c(q,D)]

A capacity decision is evaluated against both the nominal demand distribution and the worst probability redistribution inside a total-variation ambiguity set. The ambiguity radius ρ controls how much probability mass an adversary may move.

Nominal capacity
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Robust capacity
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Nominal price of robustness
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Research promptsAt what ambiguity radius does the robust capacity first differ from the nominal decision?How should ρ be calibrated from held-out data rather than selected to justify a preferred answer?

This finite-support example uses a total-variation ball and a piecewise-linear capacity cost. Other ambiguity sets encode different notions of plausible distribution shift and can produce different decisions.

Research basis [7] [8] [9]

Read the full article: A Decision That Survives the Shift →

References

Sources behind the inquiry design.

Foundational papers define the inferential problems; newer methodological papers show how the research programmes continue to develop. They support the framing, not the numerical output of these synthetic browser experiments.

  1. Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmüller, U., & Timmer, J. (2009). Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics, 25(15), 1923–1929.
  2. Wieland, F.-G., Hauber, A. L., Rosenblatt, M., Tönsing, C., & Timmer, J. (2021). On structural and practical identifiability. Current Opinion in Systems Biology, 25, 60–69.
  3. Díaz-Seoane, S., Rey Barreiro, X., & Villaverde, A. F. (2023). STRIKE-GOLDD 4.0: user-friendly, efficient analysis of structural identifiability and observability. Bioinformatics, 39(1), btac748.
  4. Hudgens, M. G., & Halloran, M. E. (2008). Toward causal inference with interference. Journal of the American Statistical Association, 103(482), 832–842.
  5. Aronow, P. M., & Samii, C. (2017). Estimating average causal effects under general interference, with application to a social network experiment. The Annals of Applied Statistics, 11(4), 1912–1947.
  6. Harshaw, C., Sävje, F., Eisenstat, D., Mirrokni, V., & Pouget-Abadie, J. (2023). Design and analysis of bipartite experiments under a linear exposure-response model. Electronic Journal of Statistics, 17(1), 464–518.
  7. Wiesemann, W., Kuhn, D., & Sim, M. (2014). Distributionally robust convex optimization. Operations Research, 62(6), 1358–1376.
  8. Mohajerin Esfahani, P., & Kuhn, D. (2018). Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations. Mathematical Programming, 171, 115–166.
  9. Chen, L., Fu, C., Si, F., Sim, M., & Xiong, P. (2025). Robust optimization with moment-dispersion ambiguity. Operations Research, 73(6), 3118–3138.