DINR: Dynamical Implicit Neural Representations

Visual Overview

Static mappings become latent trajectories.

Instead of sending each coordinate through a fixed feedforward transformation, DINR evolves embedded coordinates along a learned continuous-time path before decoding the final signal value.

Static INRs map coordinates through a fixed feedforward transformation; DINR evolves latent features along a learned trajectory before decoding the signal.

DINR network design comparing a static latent mapping with dynamic latent evolution and shared vector-field updates — DINR replaces a static latent stack with a shared vector field that evolves features before decoding the output signal.

Abstract

Continuous dynamics for sharper neural fields.

Implicit Neural Representations (INRs) provide a powerful continuous framework for modeling complex visual and geometric signals, but spectral bias remains a fundamental challenge, limiting their ability to capture high-frequency details. Orthogonal to existing remedy strategies, we introduce Dynamical Implicit Neural Representations (DINR), a new INR modeling framework that treats feature evolution as a continuous-time dynamical system rather than a discrete stack of layers. This dynamical formulation mitigates spectral bias by enabling richer, more adaptive frequency representations through continuous feature evolution. Theoretical analysis based on Rademacher complexity and the Neural Tangent Kernel demonstrates that DINR enhances expressivity and improves training dynamics. Moreover, regularizing the complexity of the underlying dynamics provides a principled way to balance expressivity and generalization. Extensive experiments on image representation, field reconstruction, and data compression confirm that DINR delivers more stable convergence, higher signal fidelity, and stronger generalization than conventional static INRs.

Controlled latent dynamics for INRs.

Better expressivity and training dynamics.

Strong results across images, scientific fields, compression, and sparse 3D data.

Method

From Static Mappings to Latent Dynamics

DINR is a plug-and-play dynamical layer for coordinate-based representations. The same coordinate input becomes an evolving latent state, and the learned vector field is shared across the trajectory.

Static INR

A conventional INR maps coordinates to a signal through a single fixed-depth transformation.

coordinate embedding static latent mapping decoder

Single pass, fixed-depth transformation.

DINR

DINR turns the latent representation into an initial condition for a neural ODE, then decodes the final state.

coordinate embedding latent ODE trajectory decoder

Iterative trajectory updates with a shared vector field.

KE regularization. Kinetic-energy regularization discourages overly fast or irregular latent motion, encouraging smoother trajectories and better generalization.

Results

Experiments: Six Questions, One Pattern

Across representation quality, scientific compression, sparse 3D reconstruction, optimization behavior, regularization, and sensitivity, DINR consistently improves static INR backbones.

RQ1

Representation quality on standard benchmarks

DINR improves PSNR/SSIM across FFNet, SIREN, WIRE, and FINER backbones on DIV2K/Kodak-style image representation tasks.

+2.36 dB FINER scene 1 +1.10 dB FINER scene 2 PSNR ↑ / SSIM ↑

DIV2K and Kodak image representation results comparing INR and DINR reconstructions for FFNet, SIREN, WIRE, and FINER — Across INR backbones, DINR improves reconstruction quality and reduces residual error on image representation benchmarks.

RQ2

Compact scientific data representation

DINR reduces model size and improves compression by sharing parameters across repeated latent-dynamics evaluations, while preserving visual and spectral fidelity.

FFNet turbulence 727,297 → 465,939 params
1.44x → 2.25x compression

FFNet weather 892,417 → 599,065 params
1.16x → 1.73x compression

FINER turbulence 198,401 → 150,343 params
5.29x → 6.97x compression

Visual and spectral comparisons include prediction, error map, and 2D power spectrum.

Turbulence and weather-field compression comparison showing FFNet and Dynamical FFNet predictions, prediction errors, and two-dimensional power spectra — The 2D power spectra show that DINR better preserves high-frequency energy than static INR baselines, helping explain its improved fidelity on turbulence and weather-field compression.

RQ3

Sparse 3D reconstruction

Cryo-EM Dynamical FFNet improves the FSC=0.5 cutoff from 0.425 to 0.587.

Cloud Dynamical variants better preserve filamentary, localized 3D structures.

Pattern Latent dynamics help reconstruct missing structure from sparse samples.

Fourier shell correlation curves showing Dynamical FFNet reaches the FSC 0.5 threshold at 0.587 compared with 0.425 for FFNet — FSC analysis shows a higher spatial-frequency cutoff for Dynamical FFNet.

Sparse cloud reconstruction comparing ground truth, FFNet, Dynamical FFNet, SIREN, and Dynamical SIREN volumes — Dynamical variants better preserve localized cloud structures than their static counterparts.

RQ4

Training dynamics and NTK

Faster convergence Latent dynamics accelerate optimization.

Higher effective NTK rank DINR broadens the active training directions.

Better conditioning Lower condition number and broader eigenspectrum.

Training error curves showing Dynamical FFNet converges faster and lower than FFNet on turbulence, Kodak, and DIV2K — Dynamical FFNet reaches lower error more quickly across turbulence, Kodak, and DIV2K training runs.

FFNet NTK condition number comparison showing DINR maintains a much lower condition number over training — FFNet condition number remains far lower with latent dynamics.

SIREN NTK condition number comparison showing lower conditioning for the dynamical variant — SIREN also benefits from better NTK conditioning under DINR.

FFNet effective rank comparison showing Dynamical FFNet maintains a much higher effective NTK rank — Dynamical FFNet keeps many more effective NTK directions active.

SIREN effective rank comparison showing the dynamical variant has a higher effective rank — Dynamical SIREN similarly preserves a higher effective rank.

Representative FFNet NTK eigenvalue distribution showing a broader eigenspectrum for Dynamical FFNet — A representative final-epoch eigenspectrum shows broader active eigenvalue support for the dynamical model.

RQ5

KE regularization ablation

Noise robustness KE lowers FFNet error by up to 18.3%.

Scarce data With SIREN and only 25% data, relative error drops by 64.8%.

Smoother dynamics The latent path stays expressive without becoming irregular.

Noise ablation box plots comparing static, dynamical without KE, and dynamical with KE regularization for FFNet and SIREN — KE regularization improves robustness as the input noise level increases.

Scarce data ablation box plots comparing static, dynamical without KE, and dynamical with KE regularization for FFNet and SIREN — With less training data, KE-regularized DINR lowers relative error for both FFNet and SIREN.

RQ6

Sensitivity analysis

More ODE integration steps generally improve performance; overly strong KE regularization can over-constrain the dynamics.

Sensitivity heatmap showing loss across ODE integration steps and kinetic energy regularization strengths for Dynamical FFNet — Performance improves with more integration steps, while very strong kinetic-energy regularization can over-constrain the latent dynamics.

Qualitative Comparison

The training animations show the same baseline backbones before and after adding dynamical latent evolution. Watch for sharper high-frequency details and more faithful reconstruction in the dynamical variants.

Animated FFNet reconstruction comparison — Static FFNet provides a compact coordinate mapping but can smooth fine detail.

Animated Dynamical FFNet reconstruction comparison — DINR adds latent evolution, improving detail recovery without changing the viewing task.

Animated SIREN reconstruction comparison — Static periodic activations capture structure, but optimization can remain uneven.

Animated Dynamical SIREN reconstruction comparison — The dynamical variant encourages richer feature evolution and stronger fidelity.

Limitations and Future Directions

DINR has higher per-iteration cost because the shared dynamical module is evaluated repeatedly along the latent trajectory. Future work can explore adaptive step-size solvers, higher-order integration, and more efficient implementations, while extending the framework to large-scale and multimodal datasets.

BibTeX

@article{park2025dynamical,
  title={Dynamical Implicit Neural Representations},
  author={Park, Yesom and Kan, Kelvin and Flynn, Thomas and Huang, Yi and Yoo, Shinjae and Osher, Stanley and Luo, Xihaier},
  journal={arXiv preprint arXiv:2511.21787},
  year={2025}
}