Towards Identifiability of Interventional Stochastic Differential Equations

Aaron Zweig; David Knowles; Elham Azizi; Zaikang Lin

REVIEW 3 major objections 4 minor 16 references

Towards Identifiability of Interventional Stochastic Differential Equations

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that a stochastic differential equation's drift can be recovered from stationary snapshots alone when the number of shift interventions reaches the hidden rank: r interventions for linear systems, r+1 for nonlinear…

desk verdict Novel intervention-count bounds for SDE identifiability, but the linear upper-bound proof has a false algebraic identity; worth a careful referee. read the letter →

arxiv 2505.15987 v5 pith:QNJ7GR43 submitted 2025-05-21 cs.LG

Aaron Zweig , Zaikang Lin , Elham Azizi , David Knowles This is my paper

classification cs.LG MSC 60H1060J60

keywords stochasticdifferentialequationsidentifiabilityinterventionsstationarydistributionshiftlow-rankdynamicsLyapunovequationgeneregulatorynetworks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how many shift interventions are needed to recover the drift of a stochastic differential equation when only stationary snapshots are observed, not trajectories. The main answer is that the number scales with the hidden rank of the dynamics, not the ambient dimension: for linear SDEs with drift $(AB-D)x$, $r$ interventions identify the drift almost surely under generic parameter samples, and at most $r-2$ interventions almost surely do not. For nonlinear SDEs of the form $A\sigma(Bx)-x$, the paper proves that $r+1$ interventions recover $A$ and $B$ up to permutation and scaling in the small-noise limit, even when the elementwise activation $\sigma$ is unknown. These are the first provable intervention counts for SDE parameter recovery from stationary data, and they are validated in synthetic and semi-synthetic gene-regulatory experiments. If correct, they justify inferring gene regulatory networks from pooled single-cell perturbation screens without temporal measurement.

What carries the argument

The load-bearing identity is the Lyapunov equation $L\omega + \omega L^\top + I = 0$, which ties the stationary covariance $\omega$ to the drift Jacobian $L$ at the fixed point. In the linear case it gives an essentially closed-form map from parameters to Gaussian stationary distributions; in the small-noise limit it linearizes the nonlinear SDE around each intervened fixed point. The proof then isolates the column space of $A$ from the intervention means, and uses linear combinations of rescaled covariances to extract rank-one outer products of columns of $A$ and rows of $B$, with the unknown activation cancelled by ratios of its derivatives evaluated at the fixed points. Lemma A.7 is the combinatorial core: from matrices $X D_i Y$ with diagonal $D_i$ whose diagonals are linearly independent, $X$ and $Y$ are recoverable up to permutation and scaling.

What would settle it

Choose two distinct elementwise activations $\sigma_1$ and $\sigma_2$ satisfying Assumption 4.5, sample $r+1$ interventions for a small $n$ and $r$, and compute the stationary means and rescaled covariances in the limit $\epsilon \to 0$; the theorem predicts the two models cannot produce identical moments, so any pair that does would refute the identifiability claim.

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Extended reading notes

Core claim

The central claim is that stationary distributions alone can identify the parameters of an interventional SDE, provided the number of interventions reaches the hidden rank of the drift. For the linear SDE $dX_t = ((AB-D)X_t + c_i)\,dt + \sqrt{\epsilon}\,dB_t$ with $\|A\|,\|B\| \le \gamma < 1$ and observed $D$, the drift $AB-D$ is almost surely identifiable from the stationary means and covariance under $r$ shift interventions, and almost surely unidentifiable with at most $r-2$ interventions (Theorem 4.4). For the nonlinear drift $v(x) = A\sigma(Bx) - x$ with contractive elementwise $\sigma$ satisfying Assumption 4.5, in the zero-noise limit $r+1$ interventions identify $A$ and $B$ up to simultaneous permutation and scaling of columns of $A$ and rows of $B$, without knowing $\sigma$ (Theorem 4.8). The nonlinear result rests on a perturbation bound (Theorem 4.7) showing that the stationary mean lies near the fixed point and the rescaled covariance near the Lyapunov solution of the linearized drift, so the covariance itself carries the low-rank structure across interventions.

Load-bearing premise

For the linear result, the decay matrix $D$ must be observed in advance; for the nonlinear result, the guarantee holds only in the idealized zero-noise limit with exact first and second moments, not for finite noisy samples.

Editorial extensions

If this is right

Linear SDE parameter recovery is feasible from stationary data with exactly $r$ interventions, so low-rank dynamics can be learned without any trajectory information.
In the nonlinear small-noise regime, $r+1$ interventions suffice and the activation function need not be known, which licenses the use of flexible learned activations without sacrificing identifiability.
The noiseless ODE limit requires roughly $n$ interventions, so nonzero noise is what enables the low-rank shortcut through second-order moments.
The perturbation bound gives an explicit rate at which the zero-noise-limit guarantee degrades as noise grows, quantifying how far from idealization a practical finite-noise setting can stray.
In gene-regulatory applications, the results imply that stationary single-cell perturbation screens need a number of interventions proportional to the latent module count, not the gene count.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step the paper does not take is to close the one-intervention gap in the linear bound: whether $r-1$ interventions identify the drift is left open by Theorem 4.4.
The paper's hypothesis that multiple stable equilibria under one intervention give as much information as several unimodal interventions could be tested directly by fitting Gaussian mixtures to small-noise stationary distributions.
The experiments with learned decay suggest the observed-$D$ assumption may be removable; extending the proof to simultaneously recover $D$ would make the linear result fully practical.
The sharp phase transition between $r$ and $r+1$ interventions in the nonlinear experiments hints that finite-sample theory could replace the zero-noise idealization with sample-complexity bounds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper asks a good, under-studied question: how many shift interventions identify the drift of a low-rank SDE when you only see stationary distributions. The proposed answer — r interventions for linear drifts, r+1 for nonlinear small-noise drifts, with matching lower bounds — is genuinely new. I checked the related work: Dettling et al. has no interventions, Guan et al. uses temporal marginals, Rohbeck and Lorch have no identifiability theory. The nonlinear result, identifying A and B^T up to permutation and scaling without knowing the activation, is the most interesting piece. The experiments are thorough, and the GRN application is a nice motivation. The citation pattern is fair; no circularity.

The problem is the proof of the linear upper bound. The stress-test note is correct. In the proof of Theorem A.1, the Lyapunov equation gives LωP − ωDP + ϵP = 0. Right-multiplying and rearranging correctly gives ωP = L^{-1}(ωD − ϵI)P. The manuscript instead writes 2ωP = L^{-1}(ωD + ϵI)P — a spurious factor 2 and a sign error. That identity is load-bearing: it is exactly what lets the authors claim (ωD + ϵI)P has rank n−r and is transverse to the intervention columns, which recovers L^{-1}. With the corrected identity, the matrix (ωD − ϵI)P can drop rank when ϵ coincides with an eigenvalue of ωD, and Assumptions 4.2 and 4.3 do not exclude that. So Theorem 4.4 is not established as written. The lower bound and the nonlinear theorem may survive, but the abstract's 'tight bounds' for the linear case are unsupported, and the r−1 case is left open.

There is also a gap between theory and experiments: the simulations use sigmoid activations, which violate Assumption 4.5 because the derivative tends to zero, so there is no positive lower bound τ. The experiments are better described as stress tests than as validations of the stated theorem; they should be labeled accordingly.

The flaw is specific and repairable, not a sign of a wrong research direction. The question is important, and the nonlinear identifiability result, if correct after a careful check of similar algebra in its proof, is a real contribution. I would send this to a serious referee who can verify whether the corrected identity still supports the rank/transversality argument or whether an extra assumption (e.g., on the spectrum of ωD) is needed. The paper deserves that engagement, with a request to fix the linear proof and align the experiments with the assumptions.

Referee Report

3 major / 4 minor

Summary. The paper studies identifiability of SDE parameters from stationary distributions under shift interventions. For linear SDEs with drift (AB−D)x and low-rank AB, it claims tight intervention-count bounds: r shift interventions identify the drift almost surely, while at most r−2 interventions do not, assuming the decay matrix D is observed. For nonlinear drifts Aσ(Bx)−x in the small-noise limit, it claims that r+1 interventions identify A and B^T up to simultaneous permutation and scaling, without knowledge of σ. The paper also presents experiments on synthetic data and a semi-synthetic gene regulatory network task that motivate learnable activation functions. The main theoretical contribution is the first provable intervention-count analysis for SDE parameter recovery from stationary data.

Significance. If the theorems hold, this is a substantial advance: earlier work either required trajectory or generator data, assumed the sparsity pattern was known, or provided no identifiability guarantees. The linear threshold r vs. r−2 and the nonlinear small-noise identifiability without knowing σ are novel and practically relevant. The manuscript contains full proofs, explicit assumptions, and a fairly extensive experimental pipeline, which is a strength. However, the proof of the central linear upper bound contains a false algebraic identity, and the proof of the nonlinear perturbation estimate contains an unsupported matrix-norm bound; neither issue is merely cosmetic. The nonlinear identifiability theorem appears more likely to be repairable, but the linear claim is not established as written.

major comments (3)

[Appendix A, proof of Theorem A.1, Eq. (18)] The displayed rearrangement is algebraically false. From Eq. (17), LωP = (ωD − ϵI)P, so multiplying by L^{-1} gives ωP = L^{-1}(ωD − ϵI)P, not 2ωP = L^{-1}(ωD + ϵI)P. The factor 2 and the sign change are exactly what make the next sentence ('(ωD + ϵI)P is rank n−r ...') true; with the corrected identity, im((ωD − ϵI)P) can drop below dimension n−r when ϵ coincides with an eigenvalue of ωD, and Assumptions 4.2–4.3 do not exclude that degeneracy. Consequently the written proof does not establish the upper bound of Theorem 4.4.
[Appendix A, proof of Theorem 4.7] The claim 'By assumption, L is Hurwitz, and in fact ∥L∥ < 1−γ' is false for non-normal matrices; the hypotheses only give ∥Aσ′(x*)B∥ ≤ γ, hence ∥L∥ ≤ 1+γ. The preceding bound ∥e^{Lt}∥ ≤ e^{∥L∥t} therefore does not yield the decaying integrand used in Eqs. (73)–(76). Since L is Hurwitz with spectral abscissa at most −(1−γ), a standard bound of the form C e^{−(1−γ)t} would repair the argument, but as written the proof of the perturbation estimate is incomplete.
[Section 5.2 and Appendix B.2] The numerical validation of Theorem 4.8 uses the sigmoid as the true activation, but Assumption 4.5 condition 2 requires min_x σ_i'(x) = τ > 0, and for the sigmoid inf_x σ'(x) = 0. Similarly, the activations in Appendix B.3 violate conditions 1 and 2 of Assumption 4.5. This does not affect the validity of the theorems, but it means the experiments do not actually run under the stated hypotheses; the authors should either relax the assumptions (for example, to σ' > 0 with suitable growth) or change the simulated activations so that the numerical validation matches the theory.

minor comments (4)

[Lemma A.3] The lemma states 'Let ∥σ∥∞ = γ < 1', but the proof uses the Lipschitz constant of σ; to be consistent with Assumption 4.5, this should be sup σ' = γ rather than ∥σ∥∞.
[Theorem 4.4 statement] The theorem statement omits Assumption 4.3, although the proof of Theorem A.1 uses D^{-1} explicitly via Eq. (15). The theorem should list the observed-decay assumption among its hypotheses.
[Section 5.1, Eq. (8)] The loss L_lin uses the true covariance ω in the Lyapunov term; the text should clarify that this is a population-level loss used to verify identifiability, not a finite-sample estimator.
[Appendix B.4] The GRN extraction described as A diag(α ◦ 1_N)B is not the same matrix product AB studied in Theorem 4.8; the relation between the extracted network and the theorem's low-rank parameterization should be stated explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the identifiability theorems are derived from observed stationary moments and standard Lyapunov/Woodbury arguments, and the only self-citation is not load-bearing.

full rationale

The paper's central claims do not reduce to their inputs by construction. In the linear case, Theorem A.1 takes the observed interventional means (Eq. 15) and the Lyapunov equation (Eqs. 16-17) as data and attempts to recover L; the decay matrix D is introduced as an explicit observation assumption (Assumption 4.3), not fitted from the target drift. Theorem A.2 establishes non-identifiability by constructing an alternate drift that provably leaves the means and covariance unchanged, which is the standard indistinguishability argument. The nonlinear Theorem 4.8 reduces identifiability to Lemma A.7, an algebraic statement about recovering X and Y from products XD_iY, proved independently via rank-one certificates; no fitted activation function appears in the proof, and the zero-noise-limit moments are assumed observations rather than predicted quantities. The only self-citation, Lin et al. 2025, is used for the GRN experimental architecture, not for any theorem. A possible algebraic misstep after 'Rearranging' in the proof of Theorem A.1 (the displayed identity '2ωP=L −1(ωD+ϵI)P' appears to have an incorrect factor and sign) is a proof-validity concern, not a case of a prediction being equivalent to a fit or a self-citation chain; it therefore does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central results depend on standard SDE theory plus the model assumptions listed above. No entity is invented and no number is fitted to data to make the derivations work. The most consequential axioms are the known-decay assumption for the linear theorem and the exact zero-noise-limit moment access for the nonlinear theorem.

assumptions (8)

standard math Stationary distribution exists and is unique for the SDEs considered, with density satisfying the Fokker-Planck equation.
Invoked via Theorem 2.1 from Särkkä and Solin and via Berglund; standard SDE theory.
domain assumption The shift interventions C are observed or known.
Assumption 2.2; needed to extract L^-1 C from the observed means.
domain assumption A and B are almost surely full-rank, have spectral norm smaller than one, and are rotationally invariant in distribution; each column of C has a density.
Assumption 4.2; used to guarantee range identification and generic full-rank events.
domain assumption The decay matrix D is observed in the linear case.
Assumption 4.3; the proof of Theorem 4.4 computes D^-1 C and needs D to isolate the range of A.
domain assumption The activation sigma is C^2, elementwise, with 0<tau<=sigma'<=gamma<1, |sigma''|<=M, and the set where sigma''=0 has measure zero.
Assumption 4.5; enables contraction, diffeomorphism of the fixed-point map, and generic Jacobian arguments in Lemmas A.5 and A.6.
domain assumption The hidden dimension r satisfies n > 2r.
Used in Theorem 4.8 to ensure P_A^perp B^T is full-rank and to recover B.
domain assumption In the nonlinear theorem, exact first and second moments of the stationary distribution are observed in the limit epsilon -> 0.
Theorem 4.8 is an asymptotic identifiability statement; finite-epsilon recovery is not proven.
standard math The Lyapunov equation has a unique solution and admits an integral representation.
Used in Theorem 4.7 and the perturbation argument; from Särkkä and Solin.

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Cite this review

Pith. "Pith review of Towards Identifiability of Interventional Stochastic Differential Equations." pith.science (2026). https://pith.science/paper/QNJ7GR43

@misc{pith2026250515987,
  author       = {Pith},
  title        = {Pith review of: Towards Identifiability of Interventional Stochastic Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNJ7GR43}},
  note         = {Machine review of arXiv:2505.15987}
}

read the original abstract

We study identifiability of stochastic differential equations (SDE) under multiple interventions. Our results give the first provable bounds for unique recovery of SDE parameters given samples from their stationary distributions. We give tight bounds on the number of necessary interventions for linear SDEs, and upper bounds for nonlinear SDEs in the small noise regime. We experimentally validate the recovery of true parameters in synthetic data, and motivated by our theoretical results, demonstrate the advantage of parameterizations with learnable activation functions in application to gene regulatory dynamics.

Figures

Figures reproduced from arXiv: 2505.15987 by the authors.

**Figure 2.** Normalized Frobenius error of learned drift against true drift in linear SDEs with [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Gene regulatory network recovery on three tested SDE models for 5 independent runs. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Reference graph

Works this paper leans on

16 extracted references · 10 canonical work pages

[4]

Identifying drift, diffusion, and causal structure from temporal snap- shots.arXiv preprint arXiv:2410.22729,

Vincent Guan, Joseph Janssen, Hossein Rahmani, Andrew Warren, Stephen Zhang, Elina Robeva, and Geoffrey Schiebinger. Identifying drift, diffusion, and causal structure from temporal snap- shots.arXiv preprint arXiv:2410.22729,

arXiv
[5]

Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

Diederik P Kingma. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

arXiv
[7]

Interpretable Neural ODEs for Gene Regulatory Network Discovery under Perturbations

Zaikang Lin, Sei Chang, Aaron Zweig, Elham Azizi, and David A Knowles. Interpretable neural odes for gene regulatory network discovery under perturbations.arXiv preprint arXiv:2501.02409,

work page Pith review arXiv
[9]

Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101,

I Loshchilov. Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101,

arXiv
[13]

From the Woodbury matrix identity, we see the means of all stationary distributions are conserved, ˆL−1C= (L+ωQ) −1C(19) = L−1 −L −1ω(I+QL −1ω)QL−1 C(20) =L −1C(21) Finally, note that by antisymmetry ofQ, ˆLω+ω ˆLT =Lω+ωL T +ωQω+ωQ T ω(22) =Lω+ωL T (23) which implies ˆLsatisfies the same Lyapunov equation and therefore induces the same covariance ω. PROOF...

work page 2019
[15]

With this setup, we obtain a family of distributions{ρ k}K k=0 in gene expression space

with no intervention, i.e,I 0 =∅, for a total ofK+ 1regimes. With this setup, we obtain a family of distributions{ρ k}K k=0 in gene expression space. We adapt the neural ODE architecture proposed in Lin et al. (2025). The base SDE model under Ik parametrizes the change of gene expression via the drift function as below.AandBare the coefficient matrices en...

work page 2025
[16]

Each interventional distributionρ k is obtained by taking the observational dataρ 0 as initial distribu- tion and simulating the SDE via the Euler-Maruyama method

applied to the true perturbed distribution and the samples drawn from the learned SDE. Each interventional distributionρ k is obtained by taking the observational dataρ 0 as initial distribu- tion and simulating the SDE via the Euler-Maruyama method. We fit the hyperparameters by testing the ODE architecture in Lin et al. (2025) on the same simulated data...

work page 2025
[20]

Each intervention is sampled as a Gaussian vector with mean zero and variance0.1on each entry, and from each intervened SDE we collect 5000 samples. To do this, we use the Euler-Maruyama scheme to solve the SDE, and use MCMC to draw approximately independent samples from the stationary distribution, usingdt= 0.01, a thinning factor of300and the first 500 ...

work page 2017

Show all 16 references

[2005]

Gaussian approximations of small noise diffusions in kullback-leibler divergence.arXiv preprint arXiv:1605.05878,

Daniel Sanz-Alonso and Andrew M Stuart. Gaussian approximations of small noise diffusions in kullback-leibler divergence.arXiv preprint arXiv:1605.05878,

arXiv
[2014]

Regvelo: gene-regulatory-informed dynamics of single cells.bioRxiv, pp

Weixu Wang, Zhiyuan Hu, Philipp Weiler, Sarah Mayes, Marius Lange, Jingye Wang, Zhengyuan Xue, Tatjana Sauka-Spengler, and Fabian J Theis. Regvelo: gene-regulatory-informed dynamics of single cells.bioRxiv, pp. 2024–12,

2024
[2018]

Efficient matrix sensing using rank-1 gaussian measurements

Kai Zhong, Prateek Jain, and Inderjit S Dhillon. Efficient matrix sensing using rank-1 gaussian measurements. InAlgorithmic Learning Theory: 26th International Conference, ALT 2015, Banff, AB, Canada, October 4-6, 2015, Proceedings 26, pp. 3–18. Springer,

2015
[2019]

Long-time dynamics of stochastic differential equations.arXiv preprint arXiv:2106.12998,

Nils Berglund. Long-time dynamics of stochastic differential equations.arXiv preprint arXiv:2106.12998,

arXiv
[2021]

Learning activation functions: A new paradigm for understanding neural networks.arXiv preprint arXiv:1906.09529,

Mohit Goyal, Rajan Goyal, and Brejesh Lall. Learning activation functions: A new paradigm for understanding neural networks.arXiv preprint arXiv:1906.09529,

1906 arXiv
[2022]

Scaling structural learning with no-bears to infer causal transcriptome networks

Hao-Chih Lee, Matteo Danieletto, Riccardo Miotto, Sarah T Cherng, and Joel T Dudley. Scaling structural learning with no-bears to infer causal transcriptome networks. InPacific Symposium on Biocomputing 2020, pp. 391–402. World Scientific,

2020
[2024]

The power of convex relaxation: Near-optimal matrix com- pletion.IEEE transactions on information theory, 56(5):2053–2080,

Emmanuel J Candès and Terence Tao. The power of convex relaxation: Near-optimal matrix com- pletion.IEEE transactions on information theory, 56(5):2053–2080,
[2025]

Causal modeling with stationary diffusions

Lars Lorch, Andreas Krause, and Bernhard Schölkopf. Causal modeling with stationary diffusions. InInternational Conference on Artificial Intelligence and Statistics, pp. 1927–1935. PMLR,

1927

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Identifying drift, diffusion, and causal structure from temporal snap- shots.arXiv preprint arXiv:2410.22729,

Vincent Guan, Joseph Janssen, Hossein Rahmani, Andrew Warren, Stephen Zhang, Elina Robeva, and Geoffrey Schiebinger. Identifying drift, diffusion, and causal structure from temporal snap- shots.arXiv preprint arXiv:2410.22729,

arXiv

[2] [5]

Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

Diederik P Kingma. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

arXiv

[3] [7]

Interpretable Neural ODEs for Gene Regulatory Network Discovery under Perturbations

Zaikang Lin, Sei Chang, Aaron Zweig, Elham Azizi, and David A Knowles. Interpretable neural odes for gene regulatory network discovery under perturbations.arXiv preprint arXiv:2501.02409,

work page Pith review arXiv

[4] [9]

Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101,

I Loshchilov. Decoupled weight decay regularization.arXiv preprint arXiv:1711.05101,

arXiv

[5] [13]

From the Woodbury matrix identity, we see the means of all stationary distributions are conserved, ˆL−1C= (L+ωQ) −1C(19) = L−1 −L −1ω(I+QL −1ω)QL−1 C(20) =L −1C(21) Finally, note that by antisymmetry ofQ, ˆLω+ω ˆLT =Lω+ωL T +ωQω+ωQ T ω(22) =Lω+ωL T (23) which implies ˆLsatisfies the same Lyapunov equation and therefore induces the same covariance ω. PROOF...

work page 2019

[6] [15]

With this setup, we obtain a family of distributions{ρ k}K k=0 in gene expression space

with no intervention, i.e,I 0 =∅, for a total ofK+ 1regimes. With this setup, we obtain a family of distributions{ρ k}K k=0 in gene expression space. We adapt the neural ODE architecture proposed in Lin et al. (2025). The base SDE model under Ik parametrizes the change of gene expression via the drift function as below.AandBare the coefficient matrices en...

work page 2025

[7] [16]

Each interventional distributionρ k is obtained by taking the observational dataρ 0 as initial distribu- tion and simulating the SDE via the Euler-Maruyama method

applied to the true perturbed distribution and the samples drawn from the learned SDE. Each interventional distributionρ k is obtained by taking the observational dataρ 0 as initial distribu- tion and simulating the SDE via the Euler-Maruyama method. We fit the hyperparameters by testing the ODE architecture in Lin et al. (2025) on the same simulated data...

work page 2025

[8] [20]

Each intervention is sampled as a Gaussian vector with mean zero and variance0.1on each entry, and from each intervened SDE we collect 5000 samples. To do this, we use the Euler-Maruyama scheme to solve the SDE, and use MCMC to draw approximately independent samples from the stationary distribution, usingdt= 0.01, a thinning factor of300and the first 500 ...

work page 2017