Interpolating Stochastic Differential Equations (iSDEs)

• Interpolating SDEs are stochastic processes that interpolate between endpoint distributions, enabling exact finite-time transport from a Gaussian prior to a target density.
• They employ Fokker–Planck and Hamilton–Jacobi–Bellman formulations, with forward-backward SDE representations and neural network solvers to compute potentials and scores.
• iSDEs have practical applications in generative modeling, high-dimensional sampling, speech restoration, and infinite-dimensional inverse problems, backed by theoretical performance guarantees.

Searching arXiv for papers on interpolating stochastic differential equations and stochastic interpolants. Interpolating stochastic differential equations (iSDEs) are stochastic processes constructed so that their time marginals realize a prescribed interpolation between endpoint distributions. In recent generative-model literature, they are used to bridge a Gaussian prior and a target density, or a clean signal and a degraded observation, by specifying a stochastic interpolant and then deriving a forward or reverse SDE whose marginal law matches that interpolant at each time (George et al., 1 Feb 2025, Lay et al., 10 Mar 2026). Closely related terminology also appears in infinite-dimensional Hilbert-space bridges and, in a distinct stochastic PDE literature, for equations that interpolate the stochastic heat and wave equations (Yu et al., 2 Feb 2026, Chen et al., 2021).

1. Finite-time transport by stochastic interpolants

A finite-dimensional iSDE can be built from a Gaussian prior and a target density by choosing $\pi_0=\mathcal N(0,r(0)^2 I_d)$ and $\pi_1(x)=\hat\pi(x)/Z$, together with smooth positive functions $g,r:[0,T]\to\mathbb R_+$ such that $g(0)=0$ and $g/r$ is non-decreasing. With independent random variables $x^*\sim \nu$ and $z\sim\mathcal N(0,I_d)$, the half interpolant is defined by

$x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$

Its marginal density is

$\rho(t,x):=\mathrm{Law}\{x_t\}(x),$

with $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$ and $\pi_1(x)=\hat\pi(x)/Z$0 imposed so as to determine $\pi_1(x)=\hat\pi(x)/Z$1 (George et al., 1 Feb 2025).

Writing the score as $\pi_1(x)=\hat\pi(x)/Z$2, the same construction yields a Fokker–Planck equation

$\pi_1(x)=\hat\pi(x)/Z$3

where $\pi_1(x)=\hat\pi(x)/Z$4 is arbitrary and

$\pi_1(x)=\hat\pi(x)/Z$5

By the converse of Fokker–Planck theory, the Itô SDE

$\pi_1(x)=\hat\pi(x)/Z$6

has exactly $\pi_1(x)=\hat\pi(x)/Z$7 as its marginal at time $\pi_1(x)=\hat\pi(x)/Z$8. In this formulation, simulating the SDE transports the Gaussian prior to $\pi_1(x)=\hat\pi(x)/Z$9 in time $g,r:[0,T]\to\mathbb R_+$0 provided $g,r:[0,T]\to\mathbb R_+$1 and $g,r:[0,T]\to\mathbb R_+$2 can be evaluated (George et al., 1 Feb 2025).

Within this line of work, iSDEs are characterized by two features. First, they allow a non-degenerate Gaussian prior and exact transport in finite time $g,r:[0,T]\to\mathbb R_+$3. Second, the framework generalizes the Föllmer process (Schrödinger-bridge with Dirac prior) to Gaussian priors in finite time (George et al., 1 Feb 2025).

2. Hamilton–Jacobi–Bellman structure and FBSDE formulation

The conditional expectation $g,r:[0,T]\to\mathbb R_+$4 can be expressed through a re-parameterized potential

$g,r:[0,T]\to\mathbb R_+$5

with

$g,r:[0,T]\to\mathbb R_+$6

where $g,r:[0,T]\to\mathbb R_+$7 is a design parameter ensuring well-posedness. The function $g,r:[0,T]\to\mathbb R_+$8 solves the backward nonlinear Hamilton–Jacobi–Bellman equation

$g,r:[0,T]\to\mathbb R_+$9

with terminal condition

$g(0)=0$0

Here

$g(0)=0$1

and

$g(0)=0$2

Once $g(0)=0$3 is known,

$g(0)=0$4

and $g(0)=0$5 also follows from $g(0)=0$6 (George et al., 1 Feb 2025).

This PDE admits a forward–backward stochastic differential equation representation of the form

$g(0)=0$7

with

$g(0)=0$8

By standard Pardoux–Peng theory, the solution can be read off from the FBSDE

$g(0)=0$9

$g/r$0

with the identities

$g/r$1

Accordingly, solving the FBSDE yields both $g/r$2 and $g/r$3 along forward sample paths (George et al., 1 Feb 2025).

A plausible implication is that the iSDE construction converts the transport problem into a nonlinear backward PDE whose gradient simultaneously determines the conditional bridge term and the score.

3. Learning, sampling, and theoretical guarantees

A machine-learning-based solver can parameterize $g/r$4 by a neural network $g/r$5 and compute $g/r$6 by auto-differentiation. The training procedure samples a random time $g/r$7, a state $g/r$8, and a Gaussian increment $g/r$9, then forms a one-step Euler discretization of the FBSDE:

$x^*\sim \nu$0

$x^*\sim \nu$1

$x^*\sim \nu$2

$x^*\sim \nu$3

Training minimizes the local martingale-matching loss

$x^*\sim \nu$4

together with the terminal-condition penalty

$x^*\sim \nu$5

and optimizes $x^*\sim \nu$6 by stochastic gradient descent (Adam) to drive $x^*\sim \nu$7 (George et al., 1 Feb 2025).

After training, sampling proceeds by initializing

$x^*\sim \nu$8

then iterating an Euler update in which the drift and score are re-computed from the learned $x^*\sim \nu$9:

$z\sim\mathcal N(0,I_d)$0

The details specify that $z\sim\mathcal N(0,I_d)$1 is reconstructed from $z\sim\mathcal N(0,I_d)$2, and both $z\sim\mathcal N(0,I_d)$3 and $z\sim\mathcal N(0,I_d)$4 are then evaluated from the learned potential (George et al., 1 Feb 2025).

Under smoothness conditions on $z\sim\mathcal N(0,I_d)$5, the FBSDE system admits a unique adapted solution and produces the classical solution of the HJB PDE. Relative to Path-Integral Sampler (PIS), Denosing Diffusion Sampler (DDS) and Time-Reversed Diffusion Sampler (DIS), iSDE allows a non-degenerate Gaussian prior and exact transport in finite time $z\sim\mathcal N(0,I_d)$6, and training does not require back-prop through stochastic paths (no neural SDE gradients), only one-step Euler losses. The numerical experiments report competitive performance on multimodal and high-dimensional targets, and ability to estimate normalization constants via the same learned $z\sim\mathcal N(0,I_d)$7 (George et al., 1 Feb 2025).

4. Discrete-time analysis and schedule design

A complementary line of work studies stochastic interpolants directly in discrete time. Given a base distribution $z\sim\mathcal N(0,I_d)$8, a target distribution $z\sim\mathcal N(0,I_d)$9, a coupling $x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$0 of them, a twice-differentiable interpolant $x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$1 with $x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$2 and $x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$3, and a noise-scale function $x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$4 with $x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$5, the latent process is

$x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$6

The associated continuous-time dynamics can be written as

$x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$7

with

$x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$8

$x_t = g(t)\,x^* + r(t)\,z,\qquad 0\le t\le T.$9

The Euler–Maruyama discretization is

$\rho(t,x):=\mathrm{Law}\{x_t\}(x),$0

for step sizes $\rho(t,x):=\mathrm{Law}\{x_t\}(x),$1 (Liu et al., 13 Feb 2025).

Liu et al. derive a finite-time KL bound under moment and estimator-accuracy assumptions. The main theorem states

$\rho(t,x):=\mathrm{Law}\{x_t\}(x),$2

and Taylor–Itô expansions give

$\rho(t,x):=\mathrm{Law}\{x_t\}(x),$3

If the coupling $\rho(t,x):=\mathrm{Law}\{x_t\}(x),$4 is chosen optimally, the leading discretization error scales like $\rho(t,x):=\mathrm{Law}\{x_t\}(x),$5 up to powers of $\rho(t,x):=\mathrm{Law}\{x_t\}(x),$6, and the overall KL-error bound may be expressed as

$\rho(t,x):=\mathrm{Law}\{x_t\}(x),$7

To nearly minimize $\rho(t,x):=\mathrm{Law}\{x_t\}(x),$8 subject to $\rho(t,x):=\mathrm{Law}\{x_t\}(x),$9, the proposed design sets $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$0 with $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$1, implemented as an “exponential” partition around the midpoint $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$2 (Liu et al., 13 Feb 2025).

The toy experiments compare uniform steps against exponentially decaying steps on a “checkerboard” $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$3 “spiral” example. The nonuniform schedule converges up to an order-of-magnitude faster, with 10× fewer steps to reach the same error, and additional experiments vary the base distribution to illustrate dependence on $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$4 (Liu et al., 13 Feb 2025).

5. Conditional iSDEs and fast solvers for speech restoration

In conditional diffusion models for speech restoration, the iSDE formalism is used to interpolate between a clean signal $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$5 and an observed degraded signal $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$6. The forward process is

$\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$7

with $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$8 chosen so that the marginal $\rho(0,\cdot)=\mathcal N(0,r(0)^2I_d)$9 remains Gaussian with mean

$\pi_1(x)=\hat\pi(x)/Z$00

Defining the interpolation schedule $\pi_1(x)=\hat\pi(x)/Z$01 uniquely fixes the drift:

$\pi_1(x)=\hat\pi(x)/Z$02

The reverse family of processes is

$\pi_1(x)=\hat\pi(x)/Z$03

where $\pi_1(x)=\hat\pi(x)/Z$04 gives the Probability Flow ODE and $\pi_1(x)=\hat\pi(x)/Z$05 gives the standard time-reversed SDE. Setting $\pi_1(x)=\hat\pi(x)/Z$06 recovers unconditional DPMs as a special case, while SGMSE+ uses an OU-Variance-Exploding iSDE (Lay et al., 10 Mar 2026).

The fast solver derives from splitting the reverse dynamics into a linear part

$\pi_1(x)=\hat\pi(x)/Z$07

with

$\pi_1(x)=\hat\pi(x)/Z$08

and a nonlinear correction $\pi_1(x)=\hat\pi(x)/Z$09 involving the learned score $\pi_1(x)=\hat\pi(x)/Z$10. A Taylor-series expansion of $\pi_1(x)=\hat\pi(x)/Z$11 around $\pi_1(x)=\hat\pi(x)/Z$12 yields the $\pi_1(x)=\hat\pi(x)/Z$13th-order non-linear correction

$\pi_1(x)=\hat\pi(x)/Z$14

with scalar weights $\pi_1(x)=\hat\pi(x)/Z$15 that can be precomputed in closed form for common iSDEs such as OUVE/fOUVE, or via scalar quadrature otherwise. For $\pi_1(x)=\hat\pi(x)/Z$16, the resulting 2-stage scheme “iSDE–2S–$\pi_1(x)=\hat\pi(x)/Z$17” uses 2 network-evaluations per step, so with $\pi_1(x)=\hat\pi(x)/Z$18 steps only 10 evaluations (Lay et al., 10 Mar 2026).

The implementation uses a 2D–UNet (NCSN++ backbone), denoising-score matching or $\pi_1(x)=\hat\pi(x)/Z$19-prediction loss, and an STFT-domain representation with a 32 ms Hann window and 16 ms hop. The experimental tasks are single-channel speech restoration on Noise Reduction (WHAMv2), Dereverberation (EARS-Reverb-v2), Declipping, MP3-decoding, and Bandwidth-Extension, evaluated by wideband PESQ, DistillMOS, SI-SDR, Log-Spectral Distance (LSD), and Fréchet Audio Distance (FAD). On Declipping, Dereverberation, and Noise-Reduction, iSDE–2S–0 achieves PESQ/SI-SDR/DistMOS on par with adaptive RK45 but with only 10 NFE, whereas other solvers need 40–90 NFE. On BWE and MP3 tasks, RK2(mid) and iSDE–2S–0 are roughly equivalent, and both greatly outperform Euler–Maruyama/PC at low NFE. An ablation reports that $\pi_1(x)=\hat\pi(x)/Z$20 suffices for 10-step performance, while tuning $\pi_1(x)=\hat\pi(x)/Z$21 on Noise-Reduction shows best PESQ/DistMOS at $\pi_1(x)=\hat\pi(x)/Z$22 (Lay et al., 10 Mar 2026).

6. Infinite-dimensional extensions, stable time-series variants, and distinct SPDE usage

In a separable Hilbert space $\pi_1(x)=\hat\pi(x)/Z$23, stochastic interpolants are defined by fixing two probability measures $\pi_1(x)=\hat\pi(x)/Z$24, a coupling $\pi_1(x)=\hat\pi(x)/Z$25 on $\pi_1(x)=\hat\pi(x)/Z$26, Gaussian noise $\pi_1(x)=\hat\pi(x)/Z$27 with positive-definite trace-class covariance operator $\pi_1(x)=\hat\pi(x)/Z$28, and scalar schedules $\pi_1(x)=\hat\pi(x)/Z$29 satisfying

$\pi_1(x)=\hat\pi(x)/Z$30

The interpolant is

$\pi_1(x)=\hat\pi(x)/Z$31

Its law is realized by the SDE

$\pi_1(x)=\hat\pi(x)/Z$32

with drift

$\pi_1(x)=\hat\pi(x)/Z$33

where $\pi_1(x)=\hat\pi(x)/Z$34 and $\pi_1(x)=\hat\pi(x)/Z$35 are conditional expectations. Under either a Bayesian-prior case, where the coupling has density $\pi_1(x)=\hat\pi(x)/Z$36 with $\pi_1(x)=\hat\pi(x)/Z$37 twice–Fréchet-differentiable and strongly convex, or a manifold case, where $\pi_1(x)=\hat\pi(x)/Z$38 is supported on a bounded subset of the Cameron–Martin space, the drift is Lipschitz on $\pi_1(x)=\hat\pi(x)/Z$39. Strong existence follows on $\pi_1(x)=\hat\pi(x)/Z$40, uniqueness holds under an additional independence condition on coordinates of $\pi_1(x)=\hat\pi(x)/Z$41, and the $\pi_1(x)=\hat\pi(x)/Z$42-Wasserstein gap between exact and approximate dynamics is controlled by the mean-square training errors of $\pi_1(x)=\hat\pi(x)/Z$43 and $\pi_1(x)=\hat\pi(x)/Z$44. The framework is applied to Darcy flow and Navier–Stokes vorticity inverse problems, and the abstract reports state-of-the-art results on complex PDE-based benchmarks (Yu et al., 2 Feb 2026).

A separate machine-learning usage appears in irregular time-series analysis, where Oh et al. formulate iSDEs as stable classes of Neural SDEs. The latent process

$\pi_1(x)=\hat\pi(x)/Z$45

is instantiated as a Langevin-type SDE, a Linear-Noise SDE, or a Geometric SDE. All three satisfy Lipschitz and linear-growth conditions and admit unique strong solutions; the LSDE is ergodic with a unique invariant Gibbs measure under mild dissipativity, the LNSDE admits exponential mean-square stability under sufficiently large $\pi_1(x)=\hat\pi(x)/Z$46, and the GSDE preserves nonnegativity with $\pi_1(x)=\hat\pi(x)/Z$47 absorbing and admits almost-sure exponential stability if $\pi_1(x)=\hat\pi(x)/Z$48. The observed series enters through a controlled path $\pi_1(x)=\hat\pi(x)/Z$49 and an augmented drift $\pi_1(x)=\hat\pi(x)/Z$50. Reported results include PhysioNet Mortality interpolation MSE $\pi_1(x)=\hat\pi(x)/Z$51 for Neural LSDE, PhysioNet Sepsis AUROC $\pi_1(x)=\hat\pi(x)/Z$52 for Neural LNSDE, Speech Commands accuracy $\pi_1(x)=\hat\pi(x)/Z$53 for Neural LSDE, and average accuracy about $\pi_1(x)=\hat\pi(x)/Z$54 with rank about $\pi_1(x)=\hat\pi(x)/Z$55 for Neural LNSDE over 30 UEA/UCR datasets, compared with $\pi_1(x)=\hat\pi(x)/Z$56 and rank $\pi_1(x)=\hat\pi(x)/Z$57 for a naïve Neural SDE (Oh et al., 2024).

The acronym also has a distinct meaning in stochastic PDE theory. One-parameter “interpolating” SPDEs are defined by

$\pi_1(x)=\hat\pi(x)/Z$58

with Caputo derivative order $\pi_1(x)=\hat\pi(x)/Z$59, fractional Laplacian order $\pi_1(x)=\hat\pi(x)/Z$60, and time-independent spatial Gaussian noise. Here $\pi_1(x)=\hat\pi(x)/Z$61 recovers the classical Stochastic Heat Equation and $\pi_1(x)=\hat\pi(x)/Z$62 recovers the classical Stochastic Wave Equation. Under a nonnegativity assumption on the Green function, the mild form is

$\pi_1(x)=\hat\pi(x)/Z$63

The solvability theory distinguishes a global-existence regime, a local-existence regime with blow-up at a deterministic time $\pi_1(x)=\hat\pi(x)/Z$64, and exact moment asymptotics of the form

$\pi_1(x)=\hat\pi(x)/Z$65

The critical line $\pi_1(x)=\hat\pi(x)/Z$66 produces a phase transition in the $\pi_1(x)=\hat\pi(x)/Z$67-plane, and for pure white noise the classical critical dimensions $\pi_1(x)=\hat\pi(x)/Z$68 for SHE and $\pi_1(x)=\hat\pi(x)/Z$69 for SWE emerge as special cases (Chen et al., 2021).

This suggests that “iSDE” has become an umbrella acronym rather than a single standardized object. In current arXiv usage, the dominant meaning concerns stochastic interpolant constructions for finite-time probabilistic transport and conditional generation, but the term also covers infinite-dimensional bridges, stability-oriented Neural SDE architectures for interpolation tasks, and interpolation families of SPDEs.