DeepFingers: Hybrid Operator Learning for VF

• DeepFingers is a hybrid operator-learning framework that predicts the spatiotemporal evolution of miscible viscous fingering by integrating physics-based conditioning with spectral operators.
• It combines DeepONet-style conditioning with Fourier neural operators and U-FNO layers to capture both global bulk motion and fine interfacial details, reducing nonphysical hallucinations.
• The framework outperforms baseline models by accurately reproducing key phenomena such as tip splitting, merging, and channel formation while maintaining physical plausibility.

Searching arXiv for the DeepFingers paper and the operator-learning papers it builds on. DeepFingers is a hybrid operator-learning framework for predicting the spatiotemporal evolution of miscible viscous fingering (VF) in unstable flow regimes. It was introduced in the context of a broader claim that AI models of hydrodynamically unstable transport can produce hallucinations: predictions that appear visually coherent yet are physically unrealistic, including spurious fluid interfaces and reverse diffusion that violate conservation laws (Wibawa et al., 22 Apr 2026). The framework combines the branch–trunk structure of DeepONet with FNO and U-FNO components in order to enforce what the paper describes as “balanced learning across the full spectrum of spatial modes”, with the specific learned operator

$$\mathcal{G} : (c_t, \xi) \mapsto c_{t+1}, \qquad \xi=[M,t+1].$$

Here $c_t$ and $c_{t+1}$ are concentration snapshots on a $64\times128$ grid, and $M$ is the viscosity ratio (Wibawa et al., 22 Apr 2026).

1. Origin and problem setting

DeepFingers was proposed for miscible viscous fingering, the instability that occurs when a less viscous fluid displaces a more viscous one in a porous medium or Hele-Shaw geometry (Wibawa et al., 22 Apr 2026). In the miscible setting, the interface is not a sharp material boundary but a concentration transition zone, and the paper formulates the learning problem in terms of the concentration field $c(\mathbf{x},t)$, with viscosity depending exponentially on concentration and transport governed by mass and momentum conservation (Wibawa et al., 22 Apr 2026).

The paper emphasizes that VF is difficult both numerically and statistically because the dynamics are nonlinear, nonstationary, multiscale, and highly sensitive to the viscosity ratio $M$ and to perturbations in the initial state (Wibawa et al., 22 Apr 2026). Small perturbations amplify into narrow fingers, followed by tip splitting, merging, shielding, and eventually channel formation. This makes the task more demanding than predicting a low-dimensional diagnostic: the learned object is the full spatiotemporal evolution of concentration snapshots rather than a reduced scalar observable (Wibawa et al., 22 Apr 2026).

A central premise of the work is that unstable flow exposes a failure mode in scientific ML analogous to hallucination in LLMs. The paper does not define a single scalar hallucination score. Instead, it characterizes hallucinations conceptually as predictions that are “visually coherent but physically unrealistic” or that “violate fundamental physical laws or deviate from expected flow dynamics” (Wibawa et al., 22 Apr 2026). This framing is specific to VF because visually plausible fronts can still conceal incorrect interfacial topology, mixing behavior, or transport ordering.

2. Hallucination in unstable-flow prediction

The paper identifies concrete hallucination modes in baseline models. In DAE-LSTM, early-time predictions contain spurious yellow patches of low-viscosity fluid embedded inside black regions of high-viscosity fluid; at later times, fingers become overly smooth, with diffused tips and insufficient splitting (Wibawa et al., 22 Apr 2026). In ViT, late-time predictions generate nonphysical black islands within yellow regions, violating expected spatial continuity of the displacement process (Wibawa et al., 22 Apr 2026).

The failures are not treated as merely aesthetic defects. The paper interprets them as conservation and transport violations because they imply nonphysical creation of isolated blobs or interfaces, backward or reverse mixing behavior, or transport trajectories inconsistent with advection–diffusion and monotone trend expectations (Wibawa et al., 22 Apr 2026). One example given is that, for $M=13$, baseline models can predict breakthrough too early, or even predict that breakthrough does not decrease monotonically with increasing $M$, contradicting DNS and the expected physics of stronger instability at larger viscosity contrast (Wibawa et al., 22 Apr 2026).

This analysis leads to a broader methodological claim: hallucinations in unstable flow are not simply generic prediction errors. The paper argues that they arise when the model distorts the physically relevant interaction between low-frequency bulk motion and high-frequency interfacial modes (Wibawa et al., 22 Apr 2026). A plausible implication is that visually plausible rollouts may be insufficient as a validation criterion for operator learners in instability-dominated PDE systems.

3. Spectral-bias explanation

The proposed explanation for these failures is spectral bias (Wibawa et al., 22 Apr 2026). The paper argues that standard neural architectures tend to learn an unbalanced spectrum of modes, and that this imbalance becomes especially damaging in VF because the flow contains both domain-scale structure and fine interfacial detail (Wibawa et al., 22 Apr 2026). If higher modes are over- or under-represented, the resulting rollouts may invent interfaces, smear tips, suppress tip splitting, or create isolated islands.

For diagnosis, the paper uses wavelet analysis rather than pure Fourier analysis, on the grounds that VF is nonstationary and wavelets localize in both scale and time (Wibawa et al., 22 Apr 2026). The reported spectral signatures are specific. ViT underestimates the dominant large-scale mode 1 while overestimating higher modes $2\!-\!7$, corresponding to exaggerated small-scale fluctuations and nonphysical artifacts (Wibawa et al., 22 Apr 2026). DAE-LSTM exhibits $c_t$0-dependent bias; for larger $c_t$1, higher modes $c_t$2 fall below DNS, consistent with too few fingers and overly diffused boundaries (Wibawa et al., 22 Apr 2026). By contrast, DeepFingers matches DNS much more closely across all modes, which the paper interprets as evidence of balanced learning across the full spectrum of spatial modes (Wibawa et al., 22 Apr 2026).

This spectral interpretation is the immediate rationale for the architecture. DeepFingers is not presented as a model with an explicit spectral regularization term or a formal “spectral debiasing loss.” Rather, the phrase is architectural and interpretive: the model is designed so that Fourier-domain operators capture global nonlocal dependence while U-Net-style multiscale processing restores fine interfacial structure that truncated Fourier modes alone may miss (Wibawa et al., 22 Apr 2026). This suggests that the framework’s main novelty lies in inductive bias rather than in a new optimization objective.

4. Architecture and operator formulation

DeepFingers combines DeepONet (Lu et al., 2019) with FNO (Li et al., 2020) and U-FNO components, while preserving a branch–trunk decomposition (Wibawa et al., 22 Apr 2026). The paper recalls the FNO operator form

$c_t$3

with

$c_t$4

and layer update

$c_t$5

where

$c_t$6

uses learned complex tensors on truncated Fourier modes (Wibawa et al., 22 Apr 2026). The U-FNO layer is written as

$c_t$7

with $c_t$8 a U-Net-style multiscale convolutional operator (Wibawa et al., 22 Apr 2026).

DeepONet is reviewed in the paper as learning a nonlinear operator with a branch network for the input function and a trunk network for the query variable, with output

$c_t$9

for discretized input

$c_{t+1}$0

(Wibawa et al., 22 Apr 2026). DeepFingers retains this branch–trunk logic, but the branch encoder is replaced by an FNO layer and the merged representation is refined with two U-FNO layers (Wibawa et al., 22 Apr 2026).

The architecture specified in the paper contains five stages:

Component	Input	Role
Branch network	$c_{t+1}$1	Single FNO layer with $c_{t+1}$2
Trunk network	$c_{t+1}$3	Fully connected conditioning network
Merge	branch + trunk outputs	Pointwise combination into $c_{t+1}$4 channels
Refinement	merged state	Two U-FNO layers
Projection	refined state	Map to single concentration channel

The merge operation is

$c_{t+1}$5

followed by

$c_{t+1}$6

$c_{t+1}$7

and final projection

$c_{t+1}$8

(Wibawa et al., 22 Apr 2026).

The learned operator is therefore explicitly

$c_{t+1}$9

Inputs and outputs both have shape $64\times128$0, and the merged hidden state has shape $64\times128$1 (Wibawa et al., 22 Apr 2026). Time and viscosity ratio enter only through the trunk, while the concentration field enters through the branch FNO (Wibawa et al., 22 Apr 2026). This means DeepFingers does not learn a continuous-in-time field solution directly; it learns a successor operator between concentration snapshots (Wibawa et al., 22 Apr 2026).

5. Training regime and evaluation methodology

The training data are generated by DNS simulations of VF over a range of viscosity ratios $64\times128$2, with concentration maps discretized on a $64\times128$3 grid (Wibawa et al., 22 Apr 2026). Prediction is performed auto-regressively in time: the model is initialized from a prescribed initial condition and rolled out by repeatedly feeding $64\times128$4 and $64\times128$5 to produce $64\times128$6 (Wibawa et al., 22 Apr 2026). The visible text does not provide an explicit loss formula, optimizer, learning-rate schedule, epoch count, or train/validation split, and it does not describe any conservation-law regularizer or PDE residual penalty (Wibawa et al., 22 Apr 2026). Its “physics-based” character derives from operator design and physics-based diagnostics rather than from constrained training.

The evaluation emphasizes physically meaningful behavior rather than pixelwise image error (Wibawa et al., 22 Apr 2026). The paper argues that because VF is intrinsically unstable, exact pixel alignment with DNS is not the appropriate standard. Instead it uses:

• wavelet spectral energies by mode
• domain-averaged concentration $64\times128$7
• breakthrough concentration at the outlet $64\times128$8
• degree of mixing $64\times128$9
• rate of mixing $M$0

(Wibawa et al., 22 Apr 2026)

The principal viscosity-ratio regimes explicitly discussed are

$M$1

ranging from weak to strong instability (Wibawa et al., 22 Apr 2026). The baselines are DAE-LSTM, ViT, and, in a separate mitigation experiment, a modified ViT with cross-attention inspired by CViT (Wibawa et al., 22 Apr 2026). The paper notes that cross-attention removes the black-island artifact, but still yields short and diffused fingers for $M$2, which suggests that temporal conditioning alone does not resolve the multiscale instability problem (Wibawa et al., 22 Apr 2026).

6. Reported performance, qualitative behavior, and limitations

The core aggregate RMSE table against DNS is:

Model	$M$3 RMSE	$M$4 RMSE	$M$5 RMSE
DeepFingers	0.00410	0.00966	0.00327
ViT	0.11463	0.02389	0.00523
DAE-LSTM	0.05334	0.03254	0.00385

For the mixing-rate metric, the paper reports:

• DeepFingers: $M$6 RMSE $M$7
• ViT: $M$8
• DAE-LSTM: $M$9

(Wibawa et al., 22 Apr 2026)

These numbers show DeepFingers performing best on every reported global metric (Wibawa et al., 22 Apr 2026). Qualitatively, it reproduces short slowly advancing fingers at low $c(\mathbf{x},t)$0, accelerated growth and more frequent tip splitting at higher $c(\mathbf{x},t)$1, and at $c(\mathbf{x},t)$2 even the onset of channeling (Wibawa et al., 22 Apr 2026). The paper explicitly states that it captures tip splitting, finger merging, and channel formation (Wibawa et al., 22 Apr 2026).

The evidence for reduced hallucination is both visual and metric-based. DeepFingers avoids the yellow patches found in DAE-LSTM and the black islands found in ViT (Wibawa et al., 22 Apr 2026). Spectrally, it tracks DNS mode redistribution more faithfully (Wibawa et al., 22 Apr 2026). In global transport diagnostics, it preserves the expected monotonic relationship between instability strength and breakthrough behavior (Wibawa et al., 22 Apr 2026). In uncertainty tests over varying initial conditions, its predicted PDFs for $c(\mathbf{x},t)$3, $c(\mathbf{x},t)$4, $c(\mathbf{x},t)$5, and $c(\mathbf{x},t)$6 remain close to DNS (Wibawa et al., 22 Apr 2026).

The paper is also explicit about limitations. It does not provide a rigorous quantitative hallucination score, nor an explicit conservation-penalty term in training (Wibawa et al., 22 Apr 2026). The governing PDEs and some metric definitions are deferred to supplementary material (Wibawa et al., 22 Apr 2026). The framework is trained and evaluated on a fixed $c(\mathbf{x},t)$7 grid for 2D concentration-field evolution, and generalization to different geometries, resolutions, or full coupled velocity–pressure prediction is not shown (Wibawa et al., 22 Apr 2026). Because the rollout is autoregressive, some accumulation of rollout error remains possible (Wibawa et al., 22 Apr 2026). Computational cost is not reported in the visible text, though the architecture is acknowledged to be more complex than a plain ViT or a simple latent-space recurrent model (Wibawa et al., 22 Apr 2026).

7. Position within operator learning and scientific machine learning

DeepFingers occupies a specific position within operator learning for PDE surrogates. Relative to plain FNO, it adds a DeepONet-style trunk for conditioning on regime and rollout time, and relative to plain DeepONet, it adds explicit Fourier-domain and multiscale convolutional operators in the branch and refinement stages (Wibawa et al., 22 Apr 2026). A plausible implication is that the framework is best understood not as a generic sequence predictor, but as a regime-conditioned successor operator specialized to unstable, multiscale transport.

Its broader scientific claim is that unstable flow reveals a failure mode of AI models that ordinary image realism or pointwise error metrics may miss (Wibawa et al., 22 Apr 2026). The paper’s central mechanistic thesis is that hallucinations in unstable flow emerge when model inductive bias distorts the physical spectrum of interacting scales (Wibawa et al., 22 Apr 2026). In that sense, DeepFingers is both a predictive architecture and a proposed diagnostic perspective on scientific ML.

This perspective also clarifies a common misconception. DeepFingers is not described as a physics-informed neural network in the sense of enforcing PDE residuals during training. The visible text contains no explicit conservation-law regularizer, PDE residual loss, or mass-balance constraint (Wibawa et al., 22 Apr 2026). Its physical plausibility comes instead from operator design, regime conditioning, and evaluation by physics-based observables. Another potential misconception is that the model solves hallucination through temporal conditioning alone; the cross-attention ViT experiment is presented precisely to argue that temporal conditioning helps but does not fully solve the multiscale instability problem (Wibawa et al., 22 Apr 2026).

In summary, DeepFingers is a hybrid operator learner for unstable porous-media transport that predicts

$c(\mathbf{x},t)$8

by combining DeepONet-style conditioning, FNO-based global spectral operators, and U-FNO multiscale refinement (Wibawa et al., 22 Apr 2026). Its significance lies less in a new loss function than in the claim that physical plausibility in unstable flow depends on balancing low- and high-frequency spatial modes, and that failures to do so produce hallucinations even when predicted flow fields look qualitatively plausible (Wibawa et al., 22 Apr 2026).