Fourier Latent Dynamics (FLD)

• Fourier Latent Dynamics (FLD) is a framework that encodes high-dimensional temporal data into a latent space using continuous-time Fourier parameterizations, ensuring compact and interpretable representations.
• It employs explicit sinusoidal parameterization and phase propagation to achieve reliable online control, accurate forecasting, and effective surrogate modeling across robotics, time series, and scientific applications.
• FLD offers long-horizon prediction, spectral generalization, and computational efficiency, though it requires careful hyperparameter tuning and involves complex implementation challenges.

Fourier Latent Dynamics (FLD) refers to a class of methods in which observed high-dimensional temporal or spatio-temporal data are encoded into a latent space structured by explicit, continuous-time Fourier or spectral parameterizations. This approach leverages the mathematical properties of Fourier (and related Laplace) bases to achieve compact, interpretable, and dynamics-preserving representations of systems showing inherent periodic, quasi-periodic, or multi-scale temporal behavior. FLD frameworks have been deployed in structured robot motion learning, advanced time series forecasting, surrogate modeling of scientific PDEs, and periodicity-native generative modeling for materials science. Common across these domains is the use of parameterized sinusoidal or spectral bases in the latent space, coupled with explicit propagators or state-space models that exploit the temporal coherence and regularities of the underlying process.

1. Latent Representation via Fourier Parameterization

FLD frameworks extract a latent representation from sequential data through learned encoders, with the parameters of the latent space (frequencies, amplitudes, offsets, phases) directly related to Fourier decompositions. In structured motion modeling, a segment of high-dimensional states $$s_t = (s_{t-H+1}, \dots, s_t) \in \mathbb{R}^{d \times H}$$ is mapped by a 1D convolutional encoder to a latent trajectory $z_t \in \mathbb{R}^{c \times H}$. Each channel of $z_t$ is approximated as

$$z_t^{(k)}(\tau) \approx a_k \cdot \sin(2\pi(f_k \tau + \phi_k)) + b_k$$

with parameters $a_k$ (amplitude), $f_k$ (frequency), $b_k$ (offset), and $\phi_k$ (phase), predicted by a parameterization head $p(\cdot)$ jointly trained with the rest of the model (Li et al., 2024). In scientific surrogate modeling of PDEs, spatial coordinates are embedded through trainable Fourier feature mappings like

$\Phi(x) = [\sin(2\pi B x) ; \cos(2\pi B x)]$

with matrix $z_t \in \mathbb{R}^{c \times H}$0 learned, which enables the decoder to recover high-frequency oscillations and sharp features (Salvador et al., 2024). In the context of crystal generative modeling, atomic positions are represented as truncated Fourier series of the unit-cell density (Duersch et al., 12 Feb 2026).

2. Latent Dynamics and Explicit Propagation

A core aspect of FLD is the use of explicit, analytically tractable latent evolution, often parameterized through phase advance in the Fourier basis. In motion learning, the latent phase is propagated as

$z_t \in \mathbb{R}^{c \times H}$1

while amplitude, frequency, and offset vectors are held locally constant. The full future latent (and hence the predicted state) is constructed in closed form via the inverse parameterization, avoiding black-box RNN-style rollouts and enforcing strong temporal coherence in prediction (Li et al., 2024). For surrogate PDEs, the latent state $z_t \in \mathbb{R}^{c \times H}$2 evolves through ordinary differential equations governed by liquid neural networks: $z_t \in \mathbb{R}^{c \times H}$3 or via analytically integrable circuits, and the spatial field is constructed from this latent and the Fourier-embedded coordinates (Salvador et al., 2024). In time series, Fourier filtering in the latent state-space model (SSM) gating—combined with Laplace inversion—enables explicit multi-scale, frequency-aware propagation (Zhang et al., 17 Jul 2025).

3. Training Objectives and Structural Priors

FLD systems employ training objectives that encourage self-supervised reconstruction or forecasting error minimization, often with hierarchical or multi-horizon loss decay. The canonical FLD loss for motion learning enforces multi-step consistency: $z_t \in \mathbb{R}^{c \times H}$4 where $z_t \in \mathbb{R}^{c \times H}$5 is the state reconstructed from phase-advanced latent parameters and the decoder. This enforces the sinusoidal manifold constraint and stability of frequency/amplitude/offset parameters over the prediction horizon (Li et al., 2024). In surrogate scientific ML, MSE is computed over all observed space–time points, and grid-based integration approximates the continuous field error (Salvador et al., 2024). In FLDmamba, end-to-end pointwise MSE between forecast and observed values is optimized, with additional regularization over the spectral filters and Laplace kernel parameters (Zhang et al., 17 Jul 2025).

Two central inductive biases are present: (1) explicit sinusoidal parameterization ensures each latent channel encodes trajectory segments as points on a circular or toroidal manifold, and (2) temporal (or spatio-temporal) multi-step objectives enforce long-horizon consistency, further constraining the latent dynamics to periodic trajectories (Li et al., 2024, Salvador et al., 2024).

4. Architectural Variants and Domain-Specific Instantiations

The FLD formalism has been instantiated across several research areas, each adapting the core spectral/latent machinery to the domain-specific data and tasks:

Domain	Encoder/Propagator	Decoder/Output
Robotics motion (Li et al., 2024)	1D Conv, FFT, phase-head	1D Conv, phase-based synth
Scientific ML (Salvador et al., 2024)	Liquid NN ODE/CfC, sparse	FCNN with Fourier features
Time Series (Zhang et al., 17 Jul 2025)	Mamba SSM + FFT gating	ILT-based full forecast
Crystallography (Duersch et al., 12 Feb 2026)	Complex Transformer VAE	Inverse Fourier synth

Motion modeling exploits phase propagation and skill-parameterized policy controllers. Scientific surrogates combine continuous-time latent ODEs (liquid networks) with Fourier-feature spatial decoders. FLDmamba for time series includes FMamba (Fourier-gated SSM), standard Mamba, and Laplace inversion steps. Crystallographic generative modeling encodes full periodic density via complex-valued tokens and applies latent diffusion in this spectral domain.

5. Online Control, Forecasting, and Generation

After training, FLD frameworks support various online or generative applications:

• Motion tracking and adaptation: In motion learning, a policy is conditioned on latent phase and skill parameters ($z_t \in \mathbb{R}^{c \times H}$6), enabling robust tracking of both reference and novel/interpolated motions. A fallback mechanism triggers safe latent propagation when user input is classified as out-of-distribution by the predictive loss $z_t \in \mathbb{R}^{c \times H}$7 (Li et al., 2024).
• Time series forecasting: FLDmamba models multivariate data as filtered SSM recursions, and reconstructs signals in the time domain via learned inverse Laplace transforms, capturing both periodic and transient modes even under severe measurement noise (Zhang et al., 17 Jul 2025).
• Scientific surrogate modeling: LFLDNets reconstruct the full spatio-temporal field by evaluating the FCNN decoder on arbitrary times and spatial Fourier features, bypassing numerical time-stepping and yielding rapid, differentiable surrogates for PDE dynamics (Salvador et al., 2024).
• Crystallographic generative modeling: Sampling in the latent (auxiliary) space via diffusion, followed by decoding through the VAE and inverse Fourier synthesis, produces new physically consistent, variable-multiplicity periodic crystal structures while respecting space-group symmetries (Duersch et al., 12 Feb 2026).

6. Experimental Results, Benchmarks, and Comparative Analysis

Key empirical findings across domains demonstrate the superiority and generality of FLD-style models:

• Motion representation and learning: FLD achieves strong long-horizon prediction error control (relative error $z_t \in \mathbb{R}^{c \times H}$8 over 1–2 s), smooth interpolation of skill parameters, and reliable policy tracking on unseen and interpolated motions. The fallback mechanism preserves safety under out-of-distribution conditions. Skill samplers based on ALP-GMM curriculum expand exploration and maintain higher performance under challenging conditions (e.g., high proportions of unlearnable motions) (Li et al., 2024).
• Time series forecasting: FLDmamba outperforms S-Mamba, iTransformer, Autoformer, and other baselines on standard multivariate datasets, showing 2–6% reduction in MSE across benchmarks. Its robustness to noise and long-lookback efficiency (with $z_t \in \mathbb{R}^{c \times H}$9 scaling) are highlighted (Zhang et al., 17 Jul 2025).
• Surrogate scientific modeling: LFLDNets reduce inference time by orders of magnitude compared to CPU-based solvers, with normalized MSE as low as $z_t$0 for cardiac electrophysiology. Compared to dense FC ODE-based latent models, FLD-based surrogates require fewer parameters and deliver lower error with greater stability (Salvador et al., 2024).
• Crystal generation: The truncated Fourier representation with transformer VAE and latent diffusion recovers >97% of meta-stable structures within small cells, matches coordinate-based VAE baselines for both validity and symmetry, and generalizes to variable atom multiplicities by construction (Duersch et al., 12 Feb 2026).

7. Benefits, Limitations, and Prospective Extensions

FLD offers domain-agnostic advances in model compactness, interpretability, temporal coherence, and spectral generalization. Its parameter efficiency outperforms fully-connected black-box latent models, and it provides mechanisms for arbitrary long-range forecast or generation by virtue of the closed-form propagation in the latent (phase) manifold. FFT and Laplace-based decompositions explicitly disentangle periodic, multi-scale, and transient behaviors.

Key limitations include:

• Sensitivity to hyperparameter choices (number of Fourier modes, decoder widths, RBF kernel widths, number of Laplace poles).
• Implementation complexity for end-to-end FFT/Laplace hybrid pipelines.
• Potential smoothing of fine-scale features if basis truncation is too aggressive.
• Need for retraining or parameter readjustment if data statistics or generative targets shift significantly.

Across all instantiated domains, code and recipes for reproducibility are available, facilitating practical application and further methodological extensions (e.g., physics-informed losses, domain decomposition, graph-based geometry encodings, and hybrid attention layers) (Li et al., 2024, Salvador et al., 2024, Zhang et al., 17 Jul 2025, Duersch et al., 12 Feb 2026).