DiffNMR: Diffusion-Based NMR Imaging & Reconstruction

• DiffNMR is a multifaceted term that covers both diffusion-based pore imaging and diffusion-model-based NMR reconstruction for spectra and molecular structures.
• It employs specialized gradient pulse sequences and phase recovery techniques to reconstruct arbitrary pore geometries and enhance spectral fidelity.
• Recent advances integrate denoising diffusion models and adaptive acquisition strategies, achieving significant improvements in reconstruction accuracy and structure elucidation.

DiffNMR denotes a set of NMR research programs centered on diffusion, but the term is not univocal. In one established usage, DiffNMR is a diffusion-based pore-imaging method that reconstructs the shape of arbitrary closed pores from short-gradient diffusion measurements by recovering the Fourier transform of the pore space function without a priori pore geometry (Kuder et al., 2012). In more recent arXiv usage, DiffNMR names diffusion-model-based machine-learning systems for non-uniformly sampled 2D NMR reconstruction (Yan et al., 26 May 2025), uncertainty-guided adaptive acquisition (Goffinet et al., 6 Feb 2025), NMR super-resolution across field strengths (Yan et al., 6 Feb 2025), and de novo molecular structure elucidation from NMR spectra (Yang et al., 9 Jul 2025). This suggests that “DiffNMR” has evolved from a physics-based diffusion-encoding method into a broader label for NMR inverse problems solved either by transport physics or by generative diffusion models.

1. Terminological scope and research lineages

The label has been attached to at least two distinct methodological lineages.

Usage of “DiffNMR”	Core object reconstructed	Representative papers
Diffusion-based NMR pore imaging	Closed-pore or periodic structure	(Kuder et al., 2012, Laun et al., 2013)
Diffusion-model-based NMR reconstruction	NUS spectra, higher-field-like spectra, molecular graphs	(Yan et al., 26 May 2025, Goffinet et al., 6 Feb 2025, Yan et al., 6 Feb 2025, Yang et al., 9 Jul 2025)

In the pore-imaging lineage, the governing object is the pore space function $p(\mathbf{x})$, with $p(\mathbf{x}) = 0$ outside the pore and $p(\mathbf{x}) = 1/V$ inside the pore, and the experimental strategy is to use short diffusion gradient pulses and the long-time limit so that the signal becomes directly related to the Fourier transform $p(\mathbf{q})$ (Kuder et al., 2012). In the machine-learning lineage, the underlying “diffusion” is the denoising diffusion probabilistic model or a related discrete or atomic diffusion process, with the NMR task posed as inpainting, super-resolution, or conditional structure generation (Yan et al., 26 May 2025).

A common misconception is that DiffNMR names a single standardized framework. The arXiv record does not support that interpretation. Instead, the same label has been used for physically encoded diffusion NMR, for diffusion-model-based spectrum reconstruction, and for diffusion-based molecular graph generation from spectra. A plausible implication is that the term is best understood contextually, with the surrounding equations and target variable determining its meaning.

2. DiffNMR as diffusion-based pore imaging

The foundational pore-imaging formulation reconstructs pore geometry from the diffusive motion of spins inside a closed domain. For a gradient profile $G(t)$, a diffusing particle following trajectory $\mathbf{x}(t)$ accumulates phase

$$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$

and the measured attenuation is

$S = \left\langle e^{i\phi} \right\rangle .$

The key assumptions are short diffusion gradient pulses of duration $\delta$, so that diffusion during the pulse can be neglected, and a diffusion time $T$ long enough for the spins to explore the full pore (Kuder et al., 2012).

Under equal and opposite short gradient pulses,

$p(\mathbf{x}) = 0$0

with

$p(\mathbf{x}) = 0$1

the long-time, short-pulse signal is

$p(\mathbf{x}) = 0$2

This measures only the power spectrum of the pore shape, so the phase of $p(\mathbf{x}) = 0$3 is lost and $p(\mathbf{x}) = 0$4 cannot be uniquely reconstructed from $p(\mathbf{x}) = 0$5 alone (Kuder et al., 2012). The direct consequence is that conventional two-pulse q-space encoding is insufficient for arbitrary pore-shape recovery.

The distinctive DiffNMR step is a three-pulse double wave vector sequence with gradient vectors

$p(\mathbf{x}) = 0$6

for which

$p(\mathbf{x}) = 0$7

Because this relation couples $p(\mathbf{x}) = 0$8 and $p(\mathbf{x}) = 0$9, it preserves the phase relations needed for recursive recovery of $p(\mathbf{x}) = 1/V$0 (Kuder et al., 2012). The earlier quotient

$p(\mathbf{x}) = 1/V$1

works only when $p(\mathbf{x}) = 1/V$2 is real, which occurs for point-symmetric pores satisfying $p(\mathbf{x}) = 1/V$3; the paper explicitly emphasizes that this does not hold for arbitrary pore shapes (Kuder et al., 2012).

The practical reconstruction workflow is radial and tomographic. For each gradient direction one measures $p(\mathbf{x}) = 1/V$4 over a radial set of $p(\mathbf{x}) = 1/V$5-values, optionally measures $p(\mathbf{x}) = 1/V$6 to obtain $p(\mathbf{x}) = 1/V$7, reconstructs the phase recursively, combines projections over many directions, and applies an inverse Radon transform to reconstruct the pore image (Kuder et al., 2012). The method does not require a prior model of the pore shape; the only structural assumptions are that the pore is closed, diffusion is in the long-time limit, gradient pulses are short, and the pore is fixed during acquisition.

The method was demonstrated on a cylindrical pore and an equilateral triangular pore. The direct quotient method works for the cylinder but fails for the triangle, whereas the recursive phase-recovery methods successfully reconstruct both shapes, establishing that DiffNMR can image arbitrary closed pores rather than only symmetric ones (Kuder et al., 2012).

3. Extensions to periodic and open structures

A major extension of the original pore-imaging logic is NMR-based diffusion lattice imaging for periodic open systems. That work addresses the case where particles can hop from one unit cell to another, so the closed-pore decorrelation argument no longer applies automatically. Using the SERPENT sequence—“SEquential Rephasing by Pulsed field-gradient Encoding N Time-intervals”—with three short gradient pulses satisfying

$p(\mathbf{x}) = 1/V$8

the theory shows that full structural information remains accessible for periodic open domains under a Gaussian envelope model of the long-time propagator (Laun et al., 2013).

For a periodic porous medium with pore-space indicator

$p(\mathbf{x}) = 1/V$9

the three-pulse signal can be written in terms of the diffusion propagator $p(\mathbf{q})$0. In the long-time limit and under the Gaussian envelope approximation, the signal simplifies to

$p(\mathbf{q})$1

which has the same multiplicative Fourier structure that enables pore reconstruction in closed systems (Laun et al., 2013). Two-gradient measurements provide

$p(\mathbf{q})$2

and the three-pulse signal supplies the missing phase relations. The paper treats a hexagonal lattice of cylinders, where symmetry reduces phase recovery to sign determination, and a cubic lattice of triangles, where the full complex phase must be reconstructed iteratively (Laun et al., 2013).

The forward problem for periodic media was further developed through a Bloch–Torrey analysis in a unit cell with pseudo-periodic boundary conditions. In that setting, the transverse magnetization satisfies

$p(\mathbf{q})$3

with Robin boundary condition

$p(\mathbf{q})$4

By replacing a continuous gradient profile with narrow pulses, the action of gradient and diffusion is separated, reducing the PDE in an extended periodic domain to a sequence of diffusion equations on a single unit cell (Moutal et al., 2020). This reformulation produces efficient numerics and spectral insight, including a ladder structure of complex eigenvalues, localization of eigenmodes near obstacle boundaries, branching points in the spectrum, and asymptotic regimes at low and high gradients (Moutal et al., 2020).

These developments broaden DiffNMR from a pore-imaging method for closed cavities into a more general structural spectroscopy for periodic media. They also show that phase retrieval remains central: two-pulse measurements mostly provide magnitudes, whereas multi-pulse measurements encode the phase relations needed for real-space reconstruction.

4. Transport models, inverse problems, and diffusion-aware NMR theory

The broader diffusion-NMR literature connected to DiffNMR is concerned with solving or approximating the Bloch–Torrey equation in geometrically or dynamically complex media. One direction introduces a perturbation basis for NMR diffusometry in confined porous media, representing boundaries by electrostatic charge distributions. In that approach, Dirichlet boundaries are represented by monopole-like sources and Neumann boundaries by dipole-like sources, yielding a mixed basis of bulk Fourier modes and boundary-induced surface potentials for the low-frequency part of the Laplace eigenspace (Nordin, 2013). The abstract states that the approximation error scales as

$p(\mathbf{q})$5

and the computations scale quadratically with the number of basis functions using fast multipole methods (Nordin, 2013).

Another direction treats diffusion under external potentials. For a Hookean potential

$p(\mathbf{q})$6

the NMR signal can be derived analytically by path integrals and numerically by the multiple correlation function formalism. The resulting framework retains a scalar bulk diffusivity $p(\mathbf{q})$7 and places anisotropy in the confinement tensor $p(\mathbf{q})$8, rather than in a diffusion tensor, thereby offering a distinct characterization of diffusion anisotropy (Yolcu et al., 2016). The same work interprets the Hookean model as a tractable approximation to restricted diffusion, while noting that diffraction-like features of strong restriction are not reproduced because the phase distribution remains Gaussian (Yolcu et al., 2016).

For non-Markovian fluids and confined geometries, a generalized diffusion coefficient can be constructed from molecular-dynamics pressure-tensor autocorrelation functions via analytic continuation of the Stokes–Einstein relation into the Laplace domain. The resulting program links memory kernels, viscosity, and NMR attenuation or linewidth in viscoelastic or boundary-restricted systems (Niknam et al., 2023). This suggests a version of DiffNMR in which the “diffusion coefficient” is no longer constant but geometry- and frequency-dependent.

Diffusion-aware NMR has also been generalized to coupled transport and chemistry. A concentration-weighted density-matrix formalism,

$p(\mathbf{q})$9

permits simultaneous treatment of diffusion, flow, and second-order reactions without concentration denominators, and the method is implemented in Spinach 2.11 and later (Acharya et al., 4 May 2025). The microfluidic Diels–Alder example combines reaction–diffusion–flow, a finite element model with thousands of Voronoi cells, and a spatially localised stripline radiofrequency coil (Acharya et al., 4 May 2025). Although this framework is not itself named DiffNMR in the paper title, it extends the same transport-centric view of NMR observables.

A practical experimental complement is diffusion NMR in a permanent magnetic field gradient, or STRAFI. In this setting, diffusion is encoded by a static stray-field gradient, with the excited slice thickness

$G(t)$0

The reported methodology extends accessible diffusion times from a few hundred microseconds to several tens of seconds and is demonstrated on nuclei including $G(t)$1Cl, $G(t)$2Br/$G(t)$3Br, $G(t)$4I, and $G(t)$5O, as well as on membrane porosity measurements (Suzanne et al., 9 Jan 2026). This is significant because conventional PFG-NMR is typically limited to shortest practical diffusion times of about $G(t)$6–$G(t)$7 ms in the paper’s framing (Suzanne et al., 9 Jan 2026).

5. DiffNMR as diffusion-model-based spectral reconstruction and acquisition

In 2025, DiffNMR was repurposed for a diffusion-model-based reconstruction framework for randomly undersampled 2D NMR signals acquired with non-uniform sampling. The problem setting is NUS in 2D NMR, where skipped evolution-time rows accelerate acquisition but create an ill-posed reconstruction problem with aliasing, spurious peaks, distorted intensities, reduced signal-to-noise, and loss of structural reliability (Yan et al., 26 May 2025). DiffNMR reframes this as an inpainting task and applies denoising diffusion probabilistic models to both time-time and time-frequency representations.

The paper compares denoising-plus-RePaint and conditioned diffusion formulations in both domains, producing the four models D-TT, D-TF, I-TT, and I-TF. Training uses the Artina / 100-protein benchmark: 1329 spectra across 2D/3D/4D, involving 100 proteins, expanded to more than 3500 samples by extracting original 2D spectra and flattening higher-dimensional spectra into 2D views (Yan et al., 26 May 2025). All DiffNMR variants outperform compressed sensing and low-rank approximation, and the clearest conclusion is that D-TF is the best model overall, with the time-frequency domain better than the time-time domain for this inpainting problem (Yan et al., 26 May 2025).

DiffNMR2 turns the diffusion reconstructor into an acquisition policy. It uses a DDPM with a UNet denoiser, Repaint-style inpainting, and row-wise uncertainty estimation from multiple stochastic reconstructions to choose which evolution-time rows to acquire next (Goffinet et al., 6 Feb 2025). The method initializes with Poisson-Gap sampling, then iterates row-wise scanning, a 1D Fourier transform along acquisition time, row inpainting, uncertainty aggregation

$G(t)$8

and acquisition of the rows with largest $G(t)$9 (Goffinet et al., 6 Feb 2025). The paper reports that 2% per step is the best trade-off among the evaluated completion percentages, and at 10% acquisition completion the guided strategies GS200 and GS400 improve MSE and hallucination ratio relative to Poisson-Gap, uniform sampling, CS, and LR. The abstract states improvements of 52.9% in reconstruction accuracy, 55.6% fewer hallucinated peaks, and 60% less time in complex NMR experiments (Goffinet et al., 6 Feb 2025).

DiffNMR3, called the Multi-Scale Super-Resolution model (MSSR), addresses a different inverse problem: reconstructing higher-field-like spectra from lower-field NMR data. It uses a conditional diffusion model built on a U-Net / Attention U-Net backbone, with the forward process

$\mathbf{x}(t)$0

and a conditional denoising objective on the noisy spectrum, low-resolution conditioning input, time step, and upscaling factor $\mathbf{x}(t)$1 (Yan et al., 6 Feb 2025). The model handles 14 upscaling factors spanning 400–900 MHz combinations within a single network. On the ARTINA dataset, the reported trends are lower MSE, higher $\mathbf{x}(t)$2, lower hallucination ratio, lower missed peak ratio in most regimes, lower peak MSE, higher peak $\mathbf{x}(t)$3, and significantly smaller peak intensity difference than the 14-model scale-specific baseline (Yan et al., 6 Feb 2025).

These three systems share a common logic: diffusion models are used not merely for denoising but as structured priors over missing or degraded NMR observations. The difference lies in the target—missing NUS rows, future acquisition locations, or high-field-like spectral resolution.

6. DiffNMR for molecular structure elucidation

A further shift in meaning appears in DiffNMR as a conditional discrete diffusion model for de novo molecular structure elucidation from NMR spectra. Here the objective is to learn

$\mathbf{x}(t)$4

where $\mathbf{x}(t)$5 is a molecular graph and $\mathbf{x}(t)$6 is the spectral condition (Yang et al., 9 Jul 2025). The framework consists of a molecular encoder, an NMR encoder, and a graph decoder trained through a three-stage program: Diff-AE pretraining of a molecular latent space and decoder, contrastive alignment between NMR and molecular representations, and conditional graph denoising during fine-tuning and inference (Yang et al., 9 Jul 2025).

The forward process corrupts node and edge types through discrete transition matrices, and the reverse process iteratively refines the entire graph jointly rather than generating it token by token. This is presented as a mechanism for ensuring global consistency and mitigating the error accumulation associated with autoregressive methods (Yang et al., 9 Jul 2025). The NMR encoder uses radial basis function encoding for continuous chemical shifts,

$\mathbf{x}(t)$7

together with separate Transformer encoders for $\mathbf{x}(t)$8H and $\mathbf{x}(t)$9C NMR and bidirectional cross-attention to fuse them (Yang et al., 9 Jul 2025).

Experiments use the Multimodal Spectroscopic Dataset with 794,403 unique molecules derived from the USPTO reaction dataset. With molecular formula included, the combined $$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$0H+$$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$1C model achieves Top-1/Top-5/Top-10 accuracies of 68.26%/75.04%/80.27% for molecules with $$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$2 heavy atoms, 67.10%/74.98%/79.58% for $$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$3, and 61.55%/70.05%/75.07% for $$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$4 (Yang et al., 9 Jul 2025). The paper also reports that RBF encoding improves Top-1 accuracy by more than 20% relative to discrete encoding, and that the combination of similarity filtering and retrieval initialization yields the best accuracy and Tanimoto similarity, especially for larger molecules (Yang et al., 9 Jul 2025).

A related but differently named method is ChefNMR, which frames NMR structure elucidation as conditional generation of 3D atomic coordinates from $$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$5H/$$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$6C spectra and chemical formula using an atomic diffusion model on a non-equivariant transformer backbone (Xiong et al., 2 Dec 2025). ChefNMR operates on simulated 1D NMR spectra for over 111,000 natural products, predicts coordinates

$$\phi = -\gamma \int_0^T \mathbf{G}(t)\cdot \mathbf{x}(t)\,dt,$$7

and reports over 65% top-10 accuracy on challenging natural product compounds (Xiong et al., 2 Dec 2025). The relationship between DiffNMR and ChefNMR is methodological rather than terminological: both cast NMR interpretation as conditional generative diffusion, but one generates molecular graphs and the other 3D atomic structures.

Taken together, these papers indicate that DiffNMR has come to name not only diffusion-encoded NMR physics but also a family of generative inference systems in which diffusion processes—continuous, discrete, or atomic—serve as priors for reconstructing spectra, guiding acquisition, or elucidating molecular structure.