Self-Supervised Physics-Informed Neural Networks

• Self-supervised physics-informed neural networks combine a deep encoder with a fixed, physics-based decoder to convert raw data into interpretable physical parameters like galaxy profile metrics.
• They employ self-supervision by minimizing reconstruction loss through physics-constrained objectives, enabling robust parameter estimation without the need for ground-truth labels.
• Empirical evaluations on synthetic galaxy images show precise parameter estimation with residual standard deviations of σ_A ≈ 0.00055, σ_e ≈ 0.0091, and σ_θ ≈ 0.92°, highlighting enhanced noise-tolerance and interpretability.

A self-supervised @@@@1@@@@ (SS-PINN) is a hybrid learning framework that integrates neural representations with explicit or numerically encoded physical models, resulting in architectures and training regimes that enforce physical constraints through self-generated supervision. These networks are specifically designed to address scenarios where ground-truth model parameters or exhaustive labeled data are unavailable, but where both domain knowledge and differentiable models of the governing physics are accessible. The SS-PINN paradigm yields architectures that not only reproduce observations but also maintain semantic integrity by mapping internal representations directly to interpretable parameters of physical models or governing equations. This approach contrasts with purely supervised machine learning techniques or unsupervised autoencoders, as it preserves interpretability and physical fidelity while allowing for robust, noise-tolerant parameter estimation and forward modeling.

1. Architectural Frameworks and Semantic Encoding

A typical self-supervised physics-informed neural network as described in (Aragon-Calvo, 2019) is architected as a hybrid system, frequently termed a “semantic autoencoder.” This consists of two principal components: a deep neural encoder followed by a hard-coded, non-trainable physics-based decoder.

• Encoder Design: The encoder is responsible for extracting a low-dimensional representation from input data (e.g., images), often via multiple convolutional and dense layers. For galaxy model fitting, the encoder employs three convolutional blocks (with respective kernel numbers of 32, 64, and 128), each block including two convolutional layers with ReLU activations and subsequent max-pooling. The flattened feature vector then passes through several dense layers, culminating in a final layer whose neuron outputs correspond directly to the semantic parameters of the underlying analytic or numerical model (e.g., major semi-axis $A$, ellipticity $e$, and position angle $\theta$ in an exponential light profile scenario).
• Physics-Aware Decoder: Unlike standard neural decoders, the SS-PINN utilizes a fixed, differentiable analytic layer corresponding to the physical model. For the exponential profile, the intensity $I(r)$ at each spatial coordinate is computed by

$$I(r) = I_0 \exp(-r'), \qquad r' = \sqrt{(x'/A)^2 + (y'/B)^2}, \qquad e = 1 - B/A,$$

with rotated coordinates $x', y'$ incorporating the estimated $\theta$. The decoder uses explicit equations to reconstruct observable data from the estimated parameters.

• Gradient Flow and Differentiability: All components of the model, including the physics-based decoder, must be implemented in a framework (e.g., TensorFlow) that supports automatic differentiation, ensuring that end-to-end training is feasible with stochastic gradient-based optimization.

2. Self-Supervision via Physics-Constrained Loss

The distinctive learning mechanism of SS-PINN is the absence of ground-truth parameter labels. The network supervises itself by requiring that the physics-based reconstructions generated from its latent representation match the observed data, typically through a data fidelity loss (e.g., mean absolute error between input and reconstructed output).

• Indirect Supervision: The training process minimizes the loss function with respect to the discrepancy between the original observation and the model-generated reconstruction; supervision of the semantic code (internal parameters) is achieved only via this feedback, not by direct comparison to actual parameter values.
• Semantic Bottleneck: The network’s bottleneck layer is structured such that its outputs represent physically meaningful quantities, forcing the latent code to be both interpretable and constrained by the domain physics.
• Robustness and Denoising: Since the decoder reconstructs observations exclusively based on meaningful parameters, the network implicitly acts as a denoising autoencoder—irrelevant noise in the input cannot be preserved in the reconstruction as the physics-based decoder allows only permissible variations.

3. Physical Model Integration and Parameter Interpretability

The central innovation of SS-PINN is the injection of explicit physical models into end-to-end learning systems, leading to complete semantic transparency of the learned representation. This has several consequences:

• Semantic Latent Space: The latent code is fixed to represent known model parameters, ensuring that the internal representation is directly interpretable in domain-specific terms.
• Constraint Propagation: The physical model acts as a regularizer, constraining the parameter estimation process to adhere strictly to known dynamics or symmetries.
• Estimation Performance: Empirical results demonstrate that for synthetic galaxy images, the semantic autoencoder architecture can estimate $A$, $e$, and $\theta$ with residual standard deviations of $\sigma_A \approx 0.00055$, $\sigma_e \approx 0.0091$, and $\sigma_\theta \approx 0.92^\circ$ over 1,000 test images with random Gaussian noise, indicating that physically meaningful constraints efficiently guide parameter extraction.

4. Comparative Analysis with Traditional and Contemporary Approaches

Self-supervised physics-informed neural networks offer unique advantages compared to both supervised and unsupervised machine learning strategies.

• Versus Supervised Learning: Traditional parameter estimation with neural networks requires labeled data (i.e., ground-truth parameters), which is often unavailable in practical scientific settings. SS-PINN bridges this gap by exploiting the presence of an explicit, differentiable forward physical model, eliminating the need for parameter annotations during training.
• Versus Unsupervised Methods: Standard autoencoders or unsupervised representation learning techniques (e.g., PCA) yield latent spaces that are not guaranteed to carry semantic or interpretable meaning. In contrast, SS-PINN architectures enforce physically relevant representations, avoiding the drawback of uninterpretable latent factors.

Approach	Parameter Labels Required	Semantic Latent Representation	Physics Constraints	Robustness to Noise
Supervised NN	Yes	Possible	Optional	Moderate
Unsupervised AE	No	No	No	Poor
SS-PINN (this)	No	Yes	Yes	Excellent

5. Application Domains and Limitations

The general approach of SS-PINN is applicable to inverse problems, parameter estimation, and forward modeling in domains where:

• The physics governing the data can be encoded in a differentiable form.
• Observations may be noisy, incomplete, or unlabeled in terms of underlying generative parameters.

Notable application cases include astrophysical model fitting (e.g., galaxy surface brightness profiles), but the technique generalizes naturally to scientific domains where analytic or numerical forward models are available.

Potential limitations include:

• Model Misspecification: The accuracy of the parameter estimates and reconstructions is contingent upon the correct choice of physical model. When the adopted model diverges from the true data-generating process, biases or inefficiencies may be introduced.
• Computational Intensity: The inclusion of a physics-based decoder, especially for complex analytic or numerical models, can increase training time due to the necessity of differentiating through the physical model.
• Local Minima and Convergence: As with all deep learning architectures, SS-PINN may encounter optimization challenges such as local minima or slow convergence, although the explicit physical regularization often provides additional stability.

6. Future Outlook and Research Extensions

Extensions to the SS-PINN methodology are plausible in multiple directions:

• Complex Physics: Incorporation of more complex or coupled physical models (e.g., systems of PDEs, time-dependent dynamics) into the decoder.
• Hybrid Models: Combination with data-driven modules, probabilistic or Bayesian layers for uncertainty quantification, or hierarchical structures for multi-scale modeling.
• Generalization and Robustness: Investigation of transferability, out-of-distribution generalization, and reliability of parameter estimates under model misspecification or systematic measurement noise.

The self-supervised, physics-aware paradigm as implemented in the semantic autoencoder architecture (Aragon-Calvo, 2019) represents a foundational framework for physically interpretable, label-free parameter inference and robust forward modeling in scientific machine learning.