PhysEncoder: Physics-Informed Encoding

• PhysEncoder is a physics-informed module that systematically integrates physical principles to compress high-dimensional data while preserving key dynamical structures.
• It employs techniques like registration-based autoencoders, hard-coded boundary conditions, and differential constraints to ensure robust, interpretable, and physically faithful representations.
• PhysEncoders enable scalable surrogate and generative modeling across diverse applications such as fluid dynamics, secure communications, and multimodal sensor analysis.

A Physical Encoder (PhysEncoder) is a mathematical, algorithmic, or neural module designed to encode measurements, simulation data, or physical parameters in a way that systematically incorporates underlying physical principles, dynamical structure, symmetries, or physical constraints. Across diverse research, the PhysEncoder concept serves as a critical mechanism for dimensionality reduction, interpretable latent representation, robust surrogate modeling, and physically informed generative modeling. The overarching goal is to improve accuracy, generalizability, interpretability, and physical fidelity by structurally aligning encoded representations with canonical or empirical laws of physics.

1. Dimensionality Reduction and Physics-Directed Encoding

PhysEncoders frequently solve the core problem of compressing high-dimensional physical states, simulation snapshots, or measurement streams into a lower-dimensional latent space while preserving the dominant modes of variance that are governed by physics—rather than arbitrary features.

Physics-aware registration-based autoencoders (Mojgani et al., 2020) achieve this by learning a diffeomorphic spatio-temporal grid $F$ that registers snapshot sequences from convection-dominated PDE systems on a moving (non-uniform, parameter/time-varying) grid, concentrating dynamical features (e.g. shocks, moving fronts, rotating structures) into stationary coordinates. The objective minimizes the reconstruction error:

$$\min_{F,\mathcal{U},\mathcal{V}} \| M - F^{-1}(\mathcal{U}\mathcal{V}) \|_F^2 + \Gamma_1 + \Gamma_2$$

where $M$ is the snapshot matrix, $\mathcal{U}$ and $\mathcal{V}$ are low-rank representations on the learned grid, and $\Gamma_{1,2}$ are grid smoothness penalties.

Key implications:

• Minimization of Kolmogorov n-width: Compression is performed on a registration-manifold where a low-rank structure is revealed, yielding strong lossless reconstructions for convection-dominated systems and interpretable basis functions.
• Separation of transport and diffusion: The grid aligns with moving features, reducing translation-induced variance and isolating slower, intrinsic dynamics.
• Practical examples: Rank-1 reconstructions for rotating images and traveling shocks—outperforming POD in systems with high Kolmogorov n-width.

2. Hard-Encoded Physics in Neural Architectures

Recent PhysEncoder methodologies employ hard encoding—embedding exact initial/boundary conditions and known PDE terms directly in neural architectures, achieving rigid compliance with physical laws.

The PeRCNN framework (Rao et al., 2021, Rao et al., 2021) introduces:

• An initial state generator for upsampling noisy measurements into full-resolution fields.
• A recurrent Π-block that captures nonlinear polynomial interactions by an explicit elementwise Hadamard product across parallel convolutional layers (replacing traditional nonlinear activations):

$$\widehat{F}(u^{(k)}) = \sum_{c=1}^{N_c} W_c \left[ \prod_{l=1}^{N_l} (K_{c,l} \ast u^{(k)} + b_l) \right]$$

This design enables direct approximation of nonlinear PDE terms ($u \cdot \nabla u$, $u^2v$, etc.), offering clarity in mapping neural representations to symbolic physical expressions.

Key advantages:

• Boundary conditions are strictly enforced via physics-based padding, rather than penalization.
• Known operators (e.g., diffusion) are hard-coded using finite difference stencils in the convolution kernels.
• Robustness and extrapolation: Demonstrated by sustained accuracy on long-term Burgers’ and Gray–Scott equations under severe data scarcity and noise.

3. Thermodynamic and Information-Theoretic Foundations

A distinct theoretical strand frames PhysEncoder as an information thermodynamics encoder (Tian et al., 2021), where the encoding process is itself a thermodynamic transformation governed by nonequilibrium laws.

• The encoding system $\mathsf{X}$ couples to an external source $\mathsf{Y}$, accruing mutual information $I(\mathsf{X}_t; \mathsf{Y}_t)$ at a minimum irreversible work cost:

$$\frac{W_{\mathrm{irr}}}{kT} \geq I(\mathsf{X}_t; \mathsf{Y}_t)$$

• Internal correlations within $\mathsf{X}$ enable synergistic encoding: sub-systems can locally exceed the global thermodynamic bound, as validated computationally in Ising models and neural data.

Significance:

• Encoding efficiency depends on internal order and interactions.
• Applicability connects physical encoding to design of biological and synthetic information processing systems.

4. Physics-Informed Generative and Surrogate Modeling

PhysEncoder architectures now pervade generative modeling domains, including video generation, physical sequence interpolation, and secure communications.

• Video Generation: PhysEncoder modules extract explicit and implicit physical priors (object positions, material, gravity cues) from images and inject these as conditions guiding diffusion-based video synthesis (Ji et al., 15 Oct 2025). Reinforcement learning with human feedback (Direct Preference Optimization) optimizes the physical representations to favor physically plausible outcomes.
• GANs for Manufacturing Sequences: In PCTGAN (Futase et al., 2021), the encoder estimates latent vectors conditioned on images, timing, and physical context (stress/strain), leading to intermediate shapes in die forging that respect mechanical constraints and timing.
• Secure Communications: SIMO AE-based PhysEncoders (Mohammad et al., 30 Apr 2024) learn end-to-end transmitter/receiver mappings incorporating realistic RF hardware impairments (DAC nonlinearities, phase noise, PA distortions), optimizing for both bit error rate (BER) and physical layer security by mixing conventional and entropy-based losses.

5. Differential, Variational, and Gradient-Based Physical Encoding

PhysEncoders also encode the differential structure of data, extracting governing ODEs/PDEs and imposing them as explicit constraints on new data generation or surrogate fields.

• Differential Informed Auto-Encoder (Zhang, 24 Oct 2024): The encoder either fits a function $F(u, u_t, u_{tt}) = 0$ via a neural network or identifies the principal differential constraint via local PCA. Decoders resample domains using PINNs, guaranteeing that all generated data satisfy learned differential relationships.
• Gradient Flow Encoding (Flouris et al., 1 Dec 2024): Dispenses with a neural encoder; instead, latent codes are optimized by running gradient flows defined by ODEs:

$$\frac{dz}{dt} = -\alpha(t) \nabla_z \ell(y, D(z(t), \theta)), \quad z(0) = 0$$

Solving this ODE yields $z^*$ for each sample, ensuring explicit, precise encoding—advantageous for data efficiency and scientific fidelity.

• Variational Encoder-Decoders (Venkatasubramanian et al., 6 Dec 2024): Encoders learn probabilistic mappings from high-dimensional physical parameters (e.g. transmissivity fields) to disentangled, low-dimensional ($r \sim 50$) latent codes, regularized via KL-divergence and covariance penalties to improve independence and generative match to observable response distributions.

6. Mathematical Encoding of Observable Subalgebras

From a foundational perspective, physical encoding can be characterized as a spectrum-preserving, convex map between finite-dimensional Jordan algebras of observables (Arcos et al., 27 Feb 2025). This approach transitions from full system encoding to encoding of relevant observable subalgebras—critical for quantum simulations.

• The encoding $\gamma$ satisfies:
  • Spectrum preservation: $\mathrm{spec}[\gamma(a)] = \mathrm{spec}[a]$
  • Convexity: $$\gamma(\lambda a_1 + (1-\lambda)a_2) = \lambda \gamma(a_1) + (1-\lambda)\gamma(a_2)$$
• For fermionic systems, the encoding precisely maps CAR algebra elements (or their even subalgebras due to superselection), determining simulation resource requirements and representational structure in quantum computing.

7. Feature Extraction and Physical Awareness in Multimodal Models

PhysEncoder designs specialize in extracting physically meaningful features from raw sensor data, such as sound, and translating them into embeddings for downstream models.

• Audio Encoder in ACORN (Wang et al., 10 Jun 2025): Decomposes sound into magnitude and phase channels (using $|X(f,t)|$, $\sin(\arg(X(f,t)))$, $\cos(\arg(X(f,t)))$), processes through deep convolutional and transformer layers, and exposes LLMs to physical phenomena (Doppler shifts, DoA, multipath propagation) via large-scale audio QA datasets.
• Application domains span LOS detection, Doppler estimation, spatial audio reasoning, and safety-critical scenario discrimination in vehicles.

---

Overall, the PhysEncoder paradigm encompasses a wide variety of architectures, mathematical frameworks, and encoding strategies. Its unifying feature is the explicit integration of physical laws, structures, and constraints into the encoding layer, facilitating both robust scientific modeling and new frontiers in physics-aware AI.