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Derivative-Informed Dimension Reduction of Stochastic Differential Equations: A Path-Space Sensitivity Approach

Joanna Zou, Han Cheng Lie, Youssef Marzouk

arXiv:2610.33210v1math.NAcs.NAmath.PRstat.CO

TL;DR

Uncertainty in inferred SDE drifts propagates to path-space observables, while high-dimensional inputs make forward uncertainty quantification challenging. The paper constructs a derivative-informed drift basis from a path-space sensitivity operator and establishes conditions for its sensitivity interpretation and error bounds. In numerical examples, reduced derivative-informed representations more accurately approximate mean first hitting times and their pushforward distributions than same-rank KL approximations.

  • Problem

    Finite-data drift models carry statistical error that propagates to SDE observables, and high-dimensional drift inputs make surrogate operators for forward uncertainty quantification increasingly challenging.

    Source: §1 · pp. 1–2

  • Method

    The method builds a drift basis from the spectral decomposition of a path-space sensitivity operator involving the observable’s Fréchet derivative with respect to drift perturbations.

    Source: §1 · p. 2, §5 · p. 21

  • Results

    For mean first hitting times in double-well and Müller-Brown potentials, low-rank derivative-informed approximations more accurately reproduce the observable and its pushforward distribution than same-rank KL approximations.

    Source: §5 · p. 21

  • Takeaways & Limitations

    Leading derivative-informed modes act as observable-adapted drift features that can serve as inputs to downstream parameter-to-observable surrogate models.

    Source: §5 · p. 22

    Evaluation in higher-dimensional state spaces and extensions to rare-event or unbounded-stopping-time observables remain future directions.

    Source: §5 · p. 22

Abstract

from arXiv · show

Uncertainty in the drift of a stochastic differential equation (SDE) induces uncertainty in statistical observables defined as expected functionals on path space. We propose a derivative-informed method for identifying low-dimensional structure in the drift-to-observable map, with the aim of improving the tractability of uncertainty quantification. This approach can be viewed as a generalization of the active subspace method to infinite-dimensional input spaces and to path-space observables of SDEs. Representing uncertainty in the drift by random perturbations in a Hilbert space, we construct a path-space sensitivity operator using first-order Fr\'echet derivatives of the observable with respect to the perturbation. We establish conditions under which the leading eigenfunctions of this operator correspond to directions of maximum mean squared sensitivity of the observable, thus enabling low-dimensional approximation with certified error bounds. We further introduce a martingale control variate of the Fr\'echet derivative that reduces error in sample-based estimators of the operator. Using a Galerkin scheme for numerical discretization, we show that reduced representations of the drift from the derivative-informed basis yield more accurate approximations of mean first hitting times of SDEs than the Karhunen-Lo\`eve expansion of the same truncation order.

1. Introduction.

Uncertainty in inferred SDE drift propagates to observable statistics, while high-dimensional inputs make forward uncertainty quantification and surrogate learning difficult. The paper introduces an observable-adapted derivative-informed basis for drift reduction and develops conditions, computation, and applications for it.

1. Introduction.

  • High-dimensional drift inputs make both model inference and uncertainty quantification harder, and complicate learning surrogate operators for the drift-to-observable map.Drift models inferred from finite data carry statistical error that propagates to process statistics.Source: §1 · pp. 1–2
  • Unlike drift-reduction bases tied to specific model classes, the proposed basis adapts to the observable, addressing potential accuracy loss from observable-agnostic truncation.The cited alternatives include reduced-rank Gaussian processes and low-rank value-function tensor decompositions.Source: §1 · p. 2
  • The derivative-informed basis uses the observable’s Fréchet-derivative sensitivity operator to identify drift directions and supports certified bounds on observable approximation error.The introduction also describes the basis as data-driven and suitable for forward uncertainty quantification and sensitivity analysis.Source: §1 · p. 2
  • Derivative-informed reduction covers path-dependent statistics with random stopping times, whereas Koopman approaches described here target fixed-time statistics.The paper distinguishes parameter-space reduction, which retains the full state space, from model order reduction, which reduces state dimension.Source: §1 · pp. 2–3

1.1. Summary of contributions.

  • The analysis specifies drift-space and observable conditions for a complete Cameron–Martin basis whose eigenvalues quantify sensitivity, enabling optimal subspace approximations with certified error bounds.The contribution summary also describes Galerkin discretization and Monte Carlo estimation of the derivative and sensitivity operator.Source: §1.1 · p. 3
  • A local martingale from the Fréchet-derivative formula serves as a control variate for sensitivity-operator estimates, with known first moment and no additional computational cost.The paper’s numerical experiments estimate pushforward distributions of mean first hitting times from Gaussian drift perturbations.Source: §1.1 · p. 4

2. Background.

The background defines the perturbed SDE and its path-space observables, then reviews Gaussian Hilbert-space inputs and two approaches to dimension reduction. It characterizes derivative-informed directions by observable sensitivity and relates their subspaces to optimal approximations.

2.1. Perturbed stochastic differential equations.

  • The SDE model represents drift uncertainty through an additive perturbation field scaled by the diffusion coefficient; the unperturbed equation has a unique strong solution under the stated coefficient conditions.For admissible perturbations, the paper denotes expectation under the perturbed SDE law by E_u.Source: §2.1 · p. 4

2.1.1. Path space observables of SDEs.

  • A path-space observable is an expected path functional under the SDE law; for analysis, the stopping time is truncated at a fixed maximum time T.The paper notes that bounded stopping time is a convenient sufficient condition and matches finite-time simulation budgets.Source: §2.1.1 · pp. 4–5

2.2. Gaussian measures on Hilbert spaces.

  • A Gaussian measure on a Hilbert space has Gaussian pushforwards under continuous linear functionals; the paper’s example uses a fractional-Laplacian covariance.The Cameron–Martin space is associated with the covariance operator, and the paper notes that it has measure zero under the Gaussian measure.Source: §2.2 · p. 5

2.3.1. Karhunen-Lo`eve expansion.

  • The rank-r KL approximation projects the perturbation onto leading covariance eigenfunctions, yielding an L2(µ)-optimal reconstruction whose error is the sum of trailing eigenvalues.Its basis identifies directions of maximum process variance.Source: §2.3.1 · pp. 6–7

2.3. Dimension reduction on Hilbert spaces.

  • The background contrasts covariance-based KL reduction with derivative-informed reduction, which selects drift directions by their effect on a chosen functional.The KL expansion is a spectral basis expansion of the stochastic process.Source: §2.3.1 · p. 6, §2.3.2 · p. 7

2.3.2. Derivative-informed subspaces.

  • The derivative-informed sensitivity operator is self-adjoint, positive semidefinite, and trace-class under the stated derivative condition; its eigenvalues rank maximal mean-squared directional sensitivity in orthogonal Cameron–Martin directions.The basis also has a generalized eigenvalue formulation used for discrete construction.Source: §2.3.2 · pp. 7–8

2.3.3. Subspace approximations of functionals on the Hilbert space.

  • For a rank-r projector, the optimal measurable approximation is a conditional expectation; simply evaluating F at the projected input is optimal when F is constant along eliminated directions.The derivative-informed approximation also has a certified error bound.Source: §2.3.3 · pp. 8–9

3. Main results.

The section develops derivative-informed reduction for SDE drift perturbations using path-space observable sensitivities, then discretizes the resulting basis with finite elements. Under stated conditions, the sensitivity operator has useful spectral and approximation properties.

3.1.1. Hilbert space of drift perturbations.

  • Bounded drift perturbations in a Hilbert space ensure a unique weak SDE solution; Hm(X) with m > n/2 is a canonical choice because it embeds into bounded functions.The observable is defined as an expectation under the perturbed SDE law, and its Fréchet derivatives have an analytical form under additional assumptions.Source: §3.1.1 · p. 10, §3.1.2 · p. 10

3.1.2. Fr´echet derivative of path space observables.

  • The path-space observable’s first-order Fréchet derivative is used to form the sensitivity operator, with sufficient path-functional conditions ensuring the derivative-informed subspace is well-defined.Bounded mean first hitting time observables are given as an example satisfying the stated conditions.Source: §3.1.2 · p. 11, §3.1.3 · p. 11

3.1.3. Derivative-informed subspaces for path space observables.

  • The path-space sensitivity operator is self-adjoint, positive semidefinite, and trace-class, and its eigenfunctions rank directions by mean squared observable sensitivity with low-rank approximation error guarantees.These results hold under the stated SDE and path-functional conditions.Source: §3.1.3 · p. 11

3.2. Discretization of the dimension reduction problem.

  • Finite element subspaces provide consistent finite-dimensional projections of drift perturbations, converging in H1(X) under regular mesh refinement.The discretization represents functions through nodal coefficients and approximates their L2 inner product with a mass-matrix-weighted product.Source: §3.2 · p. 12, §3.2.1 · p. 12

3.2.1. Finite element basis.

  • The finite element basis uses continuous Lagrange functions whose nodal coefficients represent projected drift perturbations.Regular mesh refinement yields convergence of the approximation in H1(X).Source: §3.2.1 · p. 12

3.2.2. Discrete representation of the derivative-informed basis.

  • Galerkin projection gives a finite-dimensional sensitivity matrix from expected outer products of observable gradients, and a derivative control variate reduces its estimation variance and rank-r subspace error.The discrete gradient is formed with respect to finite element nodal coefficients.Source: §3.2.2 · pp. 13–14

3.3. Control variate estimation of the derivative-informed subspace.

A local martingale from the Fréchet derivative formula serves as a zero-mean control variate for estimating the sensitivity matrix. It reduces estimator variance without additional computational cost, whereas outer-expectation matrix control variates require extra SDE simulations.

3.3. Control variate estimation of the derivative-informed subspace.

  • The local martingale is a zero-mean, correlated control variate for the Fréchet derivative, reducing estimator variance at no additional computational cost.Its coefficient can be chosen to minimize estimator variance.Source: §3.3 · p. 14
  • Outer-expectation matrix control variates require additional drift samples and SDE simulations, making them less practical than the derivative-based estimator.SDE path simulation dominates the cost of computing the derivative-informed subspace.Source: §3.3 · p. 14

4. Numerical studies.

The numerical studies use Galerkin finite-element discretizations to compare rank-r derivative-informed (DI) and Karhunen–Loève (KL) approximations of mean first hitting times. Across the double-well and Müller-Brown examples, DI approximations more accurately capture the observable and its pushforward distribution, with sensitivity estimates also improved by a control variate.

4.1. Double well potential.

  • The DI spectra decay more sharply, and their leading eigenfunctions concentrate in regions relevant to the observable, unlike covariance-based KL modes.In the double well, DI variation concentrates between x = −3 and the hitting domain at x = 1; in the Müller-Brown example it concentrates near the primary well and initial condition.Source: §4.1 · pp. 15–16, Figure 1 · p. 16, §4.2 · p. 19
  • In the double well, DI projections have smaller local error on approximately (−2.2, 1), while KL minimizes global L2(µ) projection error.DI projections also neglect perturbation variation in regions rarely visited by trajectories or beyond the termination point.Source: §4.1 · p. 16, Figure 2 · p. 17
  • For every tested double-well truncation order, DI yields lower observable MSE and pushforward-distribution KS statistic than KL.At r=20, DI also more accurately captures the modes of the MFHT distribution.Source: §4.1 · p. 17, Figure 3 · p. 18
  • Beyond m ≳400 perturbation samples, observable MSE is not sensitive to sample size; ranks r > 80 depend more strongly on m.For this study, modest sample counts capture dominant low-rank structure, while larger subspaces require more samples.Source: §4.1 · pp. 17–18, Figure 4 · p. 18
  • The martingale control-variate estimator reduces sensitivity-operator MSE by two orders of magnitude and also reduces DI-subspace error.The subspace error is measured by weighted subspace distance.Source: §4.1 · p. 18, Figure 5 · p. 19

4.2. M¨uller-Brown potential.

  • At r=10, DI reduces the Müller-Brown approximation from d=2998 dimensions; at low ranks, MSE and KS show up to an order-of-magnitude reduction versus KL.The DI approximation captures the reference pushforward distribution more closely; the reported error reductions apply at r ≲100.Source: §4.2 · p. 20, Figure 8 · p. 21

5. Conclusions.

The paper develops a derivative-informed basis for reducing uncertain SDE drifts and gives conditions for sensitivity interpretation and low-rank error guarantees. Numerical examples show more accurate mean first hitting time approximations than same-rank Karhunen–Loève approximations, while several extensions remain open.

5. Conclusions.

  • Under Condition 3.4, the sensitivity operator’s eigenfunctions form an orthonormal Cameron–Martin basis, its eigenvalues measure sensitivity, and its projections have low-rank error guarantees.The basis is constructed from the spectral decomposition of an operator involving the Fréchet derivative with respect to drift perturbations.Source: §5 · p. 21
  • A finite-element discretization and a martingale control variate support practical computation of the derivative-informed subspaces.The control variate has an analytical mean and negligible additional estimation cost because it is already computed in the Fréchet derivative formula.Source: §5 · p. 21
  • For mean first hitting times in double-well and Müller-Brown potentials, low-rank derivative-informed approximations better reproduce the observable and its pushforward distribution than same-rank Karhunen–Loève approximations.The modes preserve accuracy in state-space regions most relevant to the observable, including metastable-state neighborhoods, hitting domains, and boundaries.Source: §5 · p. 21
  • Leading derivative-informed modes act as observable-adapted drift features that can serve as inputs to downstream parameter-to-observable surrogate models.The modes correspond to features of the drift that govern the observable.Source: §5 · p. 22
  • The approach targets path-space observables such as first passage times and metastable-state transition rates, whose random stopping times are not directly covered by Koopman-based reduction.Numerical results indicate that these observables nevertheless have low-dimensional structure with respect to the drift.Source: §5 · p. 22
  • Proposed extensions include higher-dimensional state spaces, integration with state-space model order reduction, other uncertain inputs, and broader classes of path-space observables.The named uncertain inputs are diffusion coefficients, initial distributions, and boundary conditions; broader observables include rare events and unbounded stopping times.Source: §5 · p. 22

Appendix A. Proof of Lemma 2.9.

The proof establishes that the restricted sensitivity operator is self-adjoint, positive semidefinite, and trace-class under the stated regularity conditions.

Appendix A. Proof of Lemma 2.9.

  • Symmetry of the inner product gives ⟨K|Ew, w′⟩E = ⟨w, K|Ew′⟩E, establishing self-adjointness.The proof applies the operator’s definition to w and w′.Source: Appendix A · p. 22
  • The proof establishes that K|E is positive semidefinite.This is stated as the consequence of the preceding nonnegative quadratic-form argument.Source: Appendix A · p. 22
  • Condition 2.7 makes the restricted derivative Hilbert–Schmidt and square-integrable, yielding finite trace and hence a trace-class K|E.The proof uses the Hilbert–Schmidt inclusion E → U and the boundedness condition on DF(u).Source: Appendix A · pp. 22–23

Appendix B. Proof of Lemma 2.10.

The proof relates the Fréchet derivative restricted to Cameron–Martin directions to the full derivative, then identifies the adjoint relation used in the lemma.

Appendix B. Proof of Lemma 2.10.

  • The argument assumes a strictly positive definite covariance C and a functional F satisfying Condition 2.7.Under these assumptions the full derivative DF(u) is bounded and has an adjoint.Source: Appendix B · p. 23
  • For w in E, the restricted derivative DEF(u) is the full derivative DF(u) applied to the included direction.The proof writes DEF(u) = DF(u) ◦ ι for the inclusion map ι : E → U.Source: Appendix B · p. 23
  • The adjoint of the restricted derivative satisfies DEF(u)∗ = C DF(u)∗.The proof obtains the identity by equating the inner-product expressions for the restricted derivative and its adjoint.Source: Appendix B · p. 23

Appendix C. Proof of Lemma 2.11.

The proof establishes an orthonormal eigenbasis and shows that the ordered eigenvalues characterize the largest sensitivity available after excluding earlier eigendirections.

Appendix C. Proof of Lemma 2.11.

  • Because K|E is positive, self-adjoint, and trace-class, its eigenfunctions form an orthonormal basis of E.The eigenvalues are ordered in decreasing value.Source: Appendix C · p. 24
  • For vectors orthogonal to the first i eigenfunctions, expansion in the eigenbasis leaves only terms indexed j ≥ i.The proof uses the orthogonality constraint defining Wi.Source: Appendix C · p. 24
  • The bound is attained only by the i-th eigenfunction, up to its stated coefficient condition.The proof invokes the decreasing order of eigenvalues and specifies equality when ci = 1 and cj = 0 for j ≠ i.Source: Appendix C · p. 24

Appendix D. Proof of Proposition 3.5.

Appendix D proves Proposition 3.5 by showing that the path-space observable’s Fréchet derivative is bounded and square-integrable under the stated assumptions. This establishes continuous differentiability of the observable.

Appendix D. Proof of Proposition 3.5.

  • Bounded stopping time and the assumed square integrability of φτ make ∥DF(u)(v)∥Y finite for every v ∈ U.The proof also uses the continuous embedding of U in L∞(Rn) to bound the perturbation.Source: Appendix D · p. 24
  • The bounded linearity of DF(u) for every u implies that F is continuously differentiable, proving the first part of Condition 2.7.The boundedness result is established in Lemma D.1.Source: Appendix D · pp. 24–25
  • The proof concludes that DF(u) belongs to L2(µ, L(U, Y)) by showing the terms in the relevant integral are finite.Source: Appendix D · p. 25

Appendix E. Eigendecomposition of the covariance operator.

Appendix E states an analytic eigendecomposition for the covariance operator in a one-dimensional setting. It specifies the interval, operator, and boundary conditions for that result.

Appendix E. Eigendecomposition of the covariance operator.

  • For a one-dimensional state domain, the appendix states that the covariance operator’s eigenvalues and eigenfunctions can be written analytically.Source: Appendix E · p. 25
  • The stated eigendecomposition setup uses an interval [a, b] with length L = b − a > 0.Source: Appendix E · p. 25
  • The operator Aν = (−γ∆ + δI)ν acts on C2([a, b]) with homogeneous Neumann boundary conditions.Source: Appendix E · p. 25

Appendix F. Error metrics.

Appendix F defines metrics for observable approximation error, distribution discrepancy, estimator error, and subspace discrepancy. These metrics evaluate different aspects of the reduced representation and sensitivity-operator estimates.

Appendix F. Error metrics.

  • The L2(µ)-weighted error of a rank-r approximation Fr is referred to as the mean squared error in the path-space observable.Source: Appendix F · p. 25
  • The two-sample Kolmogorov-Smirnov statistic measures discrepancy between empirical distributions representing F♯µ and (Fr)♯µ.The empirical cumulative distribution function of (Fr)♯µ is defined from sampled reduced-observable values.Source: Appendix F · p. 25
  • The mean squared error of the sample-based estimator ˆK is measured with the Frobenius norm and admits a bias-variance decomposition.Source: Appendix F · pp. 25–26
  • The weighted subspace distance measures discrepancy between subspaces and is applied to estimates of the derivative-informed subspace.Source: Appendix F · p. 26
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