Neural Diffeomorphic Neural Operator

• NDNO is a family of operator-learning frameworks that uses smooth, invertible mappings to transform variable geometries into a common reference domain for PDE solution learning.
• It externalizes geometry via pullback and pushforward operations, enabling standard neural operators to operate on uniform data representations and reducing data requirements.
• NDNO has demonstrated improved accuracy and computational efficiency across applications like fluid dynamics and mechanics compared to traditional fixed-domain approaches.

Neural Diffeomorphic–Neural Operator (NDNO) denotes a family of operator-learning frameworks that extend neural operators from a single fixed spatial domain to families of varying geometries by composing the learned operator with diffeomorphic maps to and from a common reference domain. In the 2024 formulation, this idea appears as the “diffeomorphism neural operator (DNO),” whose central move is to transform operator learning on many physical domains into operator learning on one generic domain through precomputed diffeomorphisms; subsequent work develops “diffeomorphic latent neural operator” and learned diffeomorphic-embedding variants that retain the same pullback–operator–pushforward structure while changing how the diffeomorphism and latent operator are represented (Zhao et al., 2024).

1. Definition, scope, and nomenclature

In its canonical form, NDNO addresses the problem of learning a single solution operator over a collection of smooth domains $\{\Omega_i\}_{i=1}^n$ and parameter fields $p_i(x)$, where the target is the PDE solution $u_i(x)$. Classical neural operators, including the Fourier Neural Operator (FNO) and DeepONet-style models, are described as learning on one fixed spatial domain $\Omega$; when $\Omega_i$ changes, performance does not generally transfer without retraining or extensive multi-geometry data. NDNO removes this fixed-domain restriction by introducing smooth invertible maps $\varphi_i:\Omega_i\to\Omega_{\rm ref}$, pulling all inputs and outputs back to a shared reference domain, training one latent or reference-domain operator there, and then pushing predictions forward to the original geometry (Zhao et al., 2024).

Two terminological lines are visible in the literature. One uses “diffeomorphism neural operator” for the reference-domain construction on varying domains and parameters. Another uses “diffeomorphic latent neural operator” for a data-efficient DeepONet-style formulation in which geometry and physics are encoded jointly on the reference domain. A later mechanics-focused paper uses “neural diffeomorphic-neural operator” for an end-to-end learned diffeomorphic embedding plus neural operator pipeline on 3D point clouds (Ahmad et al., 2024). Taken together, these papers suggest that NDNO names a methodological family rather than one uniquely standardized architecture.

The common premise is geometric factorization. Instead of asking a neural operator to learn shape variability implicitly, NDNO externalizes geometry into $\varphi_i$ and reserves the neural operator for the transformed field problem. This is the essential distinction between NDNO and fixed-domain operator learning.

2. Core mathematical construction

Let $\Omega_{\rm ref}$ be a fixed reference domain, such as $[0,1]^d$. For a physical domain $\Omega_i$ and parameter fields $p_i(x)$0, the pullback to the reference configuration is

$p_i(x)$1

NDNO then learns a reference-domain operator $p_i(x)$2 such that

$p_i(x)$3

Equivalently,

$p_i(x)$4

Training is posed through an empirical $p_i(x)$5-loss over $p_i(x)$6 pairs, with an exact reference-domain expression weighted by $p_i(x)$7; in practice, point-wise sampling on a uniform grid in $p_i(x)$8 together with Monte-Carlo yields an unbiased estimate (Zhao et al., 2024).

The same pullback–pushforward notation appears in the latent formulation. If $p_i(x)$9 and $u_i(x)$0, then

$u_i(x)$1

Under this coordinate change, the differential operator changes as well. For the Laplacian,

$u_i(x)$2

This makes explicit that the choice of $u_i(x)$3 influences the regularity and structure of the transformed problem, not merely its discretization (Ahmad et al., 2024).

A distinctive diagnostic introduced in the original DNO work is the Domain Diffeomorphism Similarity (DDS), defined by normalized cross-correlation of pull-back coordinate maps. Empirically, lower DDS correlates with higher prediction error, and the proposed use of DDS is to assess generalization to an unseen domain before deployment (Zhao et al., 2024).

3. Diffeomorphic mappings and architectural realizations

In the original two-stage DNO pipeline, the diffeomorphism is a preprocessing object. In 2D, $u_i(x)$4 is obtained by solving a harmonic map $u_i(x)$5 with Dirichlet boundary conditions; practically, this is a sparse linear system on a triangle mesh whose cotangent-Laplace discretization enforces conformal bijectivity. In 3D, volume parameterization, including transfinite interpolation, maps each part into a bounding cuboid with a proof of positive Jacobian ensuring smooth invertibility. After $u_i(x)$6 is computed, a uniform grid is sampled on $u_i(x)$7, pulled back through $u_i(x)$8, and used to interpolate parameter fields into a regular array. A standard FNO is then applied on the reference domain, with lifting $u_i(x)$9, spectral layers using FFT and iFFT, and projection $\Omega$0; the reported experimental setting typically used 6 Fourier layers, 4 fully-connected layers in $\Omega$1 and $\Omega$2, and grids of resolution $\Omega$3 in 2D or $\Omega$4 in 3D (Zhao et al., 2024).

The diffeomorphic latent formulation replaces the pure FNO pipeline with a DeepONet-style architecture. It combines a geometry encoder $\Omega$5, a physical-condition encoder $\Omega$6, and a spatial trunk $\Omega$7, then forms the latent solution as

$\Omega$8

The mapping $\Omega$9 may be constructed by Schwarz–Christoffel conformal maps, LDDMM, or discrete OT. A central claim of this work is that maps respecting PDE invariances can substantially reduce data requirements; the extreme case highlighted is conformal invariance of the Laplacian (Ahmad et al., 2024).

A later 3D mechanics formulation learns $\Omega_i$0 rather than precomputing it. There, $\Omega_i$1 is realized by a registration neural network acting on source and target point clouds. The total mapping loss is

$\Omega_i$2

where $\Omega_i$3 uses the Sinkhorn distance, $\Omega_i$4 penalizes folding or inversion, and $\Omega_i$5 discourages overly wiggly mappings. The reported architecture uses two DGCNN backbones, a cross-attention Transformer, and an MLP projection module, while the operator on $\Omega_i$6 is a Geo-FNO-style network with a lifting MLP, 4 Fourier layers, 12 retained modes, and channel width 64 (Liu et al., 9 Sep 2025).

4. Reported empirical performance

The original DNO paper evaluates the framework on Darcy flow, incompressible pipe flow, airfoil flow, and 3D mechanics. The reported benchmarks are summarized below (Zhao et al., 2024).

Benchmark	Setting	Reported result
Darcy flow	200 pentagon-shaped domains; test on scaled pentagons and unseen hexagons	$\Omega_i$7-error below $\Omega_i$8 on scaled pentagons; unseen hexagons $\Omega_i$9
Pipe flow	Pipe cross-sections with 2–6 baffles; inlet velocity in $\varphi_i:\Omega_i\to\Omega_{\rm ref}$0	Velocity-field $\varphi_i:\Omega_i\to\Omega_{\rm ref}$1-errors in $\varphi_i:\Omega_i\to\Omega_{\rm ref}$2; stable over temporal snapshots
Airfoil flow	5 airfoils; Reynolds $\varphi_i:\Omega_i\to\Omega_{\rm ref}$3–$\varphi_i:\Omega_i\to\Omega_{\rm ref}$4; angle-of-attack $\varphi_i:\Omega_i\to\Omega_{\rm ref}$5	Five unseen airfoil/flow conditions with $\varphi_i:\Omega_i\to\Omega_{\rm ref}$6 errors $\varphi_i:\Omega_i\to\Omega_{\rm ref}$7
3D mechanics	12 part variants; train on first 10	Test $\varphi_i:\Omega_i\to\Omega_{\rm ref}$8 error $\varphi_i:\Omega_i\to\Omega_{\rm ref}$9; last 2 variants $\varphi_i$0; new topology $\varphi_i$1

These experiments are used to support two claims: first, that one learned operator can be reused across domains after diffeomorphic transfer; second, that DDS tracks domain-shift risk, since the paper reports a strong monotonic trend between DDS and prediction error on the Darcy task (Zhao et al., 2024).

The latent-data-efficient formulation studies the 2D Laplace equation on random doubly-connected domains mapped to a fixed annulus. Using relative $\varphi_i$2 error on test domains, the reported results are 0.26% error with 80 samples for conformal NDNO without a geometry branch, 2.56% error with 400 samples for LDDMM NDNO with 10 PCA modes, and 22.4% error with 400 samples for discrete OT NDNO. The error-versus-training-size curves are reported to show that conformal NDNO is one order of magnitude more data-efficient (Ahmad et al., 2024).

The learned-embedding 3D mechanics variant reports both accuracy and speed. For frame parts, NDNO gives avg. max-error $\varphi_i$3 and RMSE $\varphi_i$4 on main-direction prediction, compared with Geo-FNO max $\varphi_i$5 and RMSE $\varphi_i$6. For C-beam parts, NDNO reports max $\varphi_i$7 and RMSE $\varphi_i$8, compared with Geo-FNO max $\varphi_i$9 and RMSE $\Omega_{\rm ref}$0. Reported inference for a new part is $\Omega_{\rm ref}$1 for registration, $\Omega_{\rm ref}$2 for operator evaluation, and $\Omega_{\rm ref}$3 overall on a single GPU, which the paper states is $\Omega_{\rm ref}$4–$\Omega_{\rm ref}$5 faster than a mesh-based FEA solve (Liu et al., 9 Sep 2025).

5. Relation to broader diffeomorphic operator learning

A broader geometric line of work uses diffeomorphisms not only to normalize varying domains, but also to represent the evolution operator itself. In “Diffeomorphic Neural Operator Learning,” the learned object is a lift

$\Omega_{\rm ref}$6

and the one-step evolution is defined by pull-back,

$\Omega_{\rm ref}$7

This converts semigroup time evolution into group composition in $\Omega_{\rm ref}$8. The reported consequences are hard enforcement of relabeling symmetry, exact preservation of masses of level-sets, extrema, Sobolev norms, and invariants of ideal transport up to interpolation error, together with non-diffusivity. On a Zalesak slotted cylinder test, the paper reports almost machine-precision transport and mass-conservation error $\Omega_{\rm ref}$9; on 2D incompressible Euler turbulence, it reports stability up to $[0,1]^d$0, enstrophy-error $[0,1]^d$1, and recovery of the $[0,1]^d$2 cascade on $[0,1]^d$3 and $[0,1]^d$4 grids (Taylor et al., 8 Aug 2025).

NeurEPDiff is closely related at the level of diffeomorphic dynamics rather than domain transfer. It learns the EPDiff solution map from an initial velocity field to a trajectory of velocities in the LDDMM/geodesic-shooting setting. Its operator is built from global integral kernels and smooth activations in a Fourier-bandlimited representation, and the paper emphasizes resolution invariance: the same trained parameters can be used at $[0,1]^d$5, $[0,1]^d$6, or $[0,1]^d$7 without retraining. Reported CPU runtimes for EPDiff alone are $[0,1]^d$8, $[0,1]^d$9, and $\Omega_i$0 for NeurEPDiff at $\Omega_i$1, $\Omega_i$2, and $\Omega_i$3, versus $\Omega_i$4, $\Omega_i$5, and $\Omega_i$6 for FLASH, and $\Omega_i$7, $\Omega_i$8, and $\Omega_i$9 for LDDMM; average Dice scores on OASIS MRI are reported as 0.866, 0.883, and 0.878 for NeurEPDiff at those resolutions (Wu et al., 2023).

These related developments situate NDNO within a larger program of geometric operator learning in which diffeomorphism structure is not incidental preprocessing, but part of the learned representation.

6. Limitations, applications, and theoretical status

The principal limitations reported for the original NDNO/DNO framework are geometric rather than purely statistical. If two domains are only weakly diffeomorphic in the sense of low DDS, prediction error may grow. High-genus shapes or topological changes break simple harmonic or volume-parameterization approaches and require more sophisticated manifold charts. The map $p_i(x)$00 must also be precomputed and stored, so extremely fine meshes raise memory demands (Zhao et al., 2024).

The same papers outline several extensions. These include incorporating group-equivariance, such as rotation or reflection, into the reference-domain operator; learning $p_i(x)$01 jointly with the operator in an end-to-end fashion; extending the method to time-dependent domains by a time-parameterized $p_i(x)$02; and combining the framework with PINN-style physics-informed losses in data-scarce regimes. The reported application areas are design optimization in aerodynamics, real-time digital twins in manufacturing, multi-physics couplings in geosciences, and high-throughput materials screening (Zhao et al., 2024).

A theoretical counterpart appears in work on Neural Shape Operator Surrogates. There, a family of admissible domains $p_i(x)$03 is generated from one bounded Lipschitz reference domain $p_i(x)$04 by an affine-parametric diffeomorphic encoder $p_i(x)$05, and the PDE is pulled back to $p_i(x)$06. Under holomorphy and uniform coercivity assumptions, the pull-backed parametric solution family is well posed, and the map $p_i(x)$07 is holomorphic. The operator surrogate is written

$p_i(x)$08

with linear encoder $p_i(x)$09, ReLU network $p_i(x)$10, and decoder $p_i(x)$11. The main expression-rate result states that for any $p_i(x)$12, there exists $p_i(x)$13 such that

$p_i(x)$14

Harbrecht and Schwab explicitly interpret this construction as instantiating the NDNO paradigm, and their results provide uniform approximation and generalization guarantees across the admissible shape family (Harbrecht et al., 20 Apr 2026).

A recurring misconception is that diffeomorphic transfer eliminates geometry dependence altogether. The reported evidence does not support that stronger claim. Rather, NDNO replaces raw geometry variation by a structured latent geometry problem: success depends on the quality, smoothness, invertibility, and PDE-compatibility of the chosen or learned diffeomorphism, and generalization remains sensitive to the transformed similarity of domains, as quantified in the original framework by DDS (Zhao et al., 2024).