Physics Informed Neural Networks

• Physics-informed neural networks (PINNs) are deep learning frameworks that incorporate differential equations as soft constraints into the training process.
• They combine data loss with physics loss through a composite loss function to ensure both data fidelity and adherence to physical laws.
• PINNs enable efficient, high-accuracy solutions for applications such as power system dynamics, parameter identification, and inverse problem solving.

Physics-informed neural networks (PINNs) constitute a framework for integrating established physical laws, typically expressed as differential equations, directly into the training of artificial neural networks. This approach is motivated by the need to augment purely data-driven models—often limited by data scarcity or noise—with the inductive bias and structural guarantees provided by physics, resulting in surrogate models that are efficient, accurate, and physically consistent. Recent research has formalized the PINN paradigm in diverse contexts, spanning power systems dynamics, continuum micromechanics, control-oriented building modeling, electromagnetic simulation, inverse scattering, and parametric design optimization.

1. Formulation and Mathematical Framework

The essential innovation in PINNs lies in embedding the governing equations of a physical system as soft constraints in the neural network’s loss function. Consider a system whose state $u(t, x)$ is governed by an operator equation

$\mathcal{N}[u; \lambda] = 0,$

with $\mathcal{N}$ a (possibly nonlinear) differential operator parametrized by system parameters $\lambda$. In PINNs, a neural network $u^\theta(t, x)$ (parameterized by $\theta$) is trained such that not only does it match observed or boundary/initial data at designated points $(t_i, x_i)$ (data loss), but it also yields a low residual $f(t, x) = \mathcal{N}[u^\theta; \lambda]$ at collocation points drawn over the domain (physics loss). The composite loss function is typically of the form:

$$\mathrm{MSE} = \frac{1}{N_u} \sum_{i=1}^{N_u} |u^\theta(t_i, x_i) - u_i|^2 + \frac{1}{N_f} \sum_{j=1}^{N_f} |f(t_j, x_j)|^2.$$

Automatic differentiation enables meshless, domain-agnostic computation of the necessary derivatives, thereby facilitating the imposition of PDEs, ODEs, or more general operator constraints within frameworks such as TensorFlow or PyTorch.

A particular instantiation for power systems dynamics involves the classic swing equation, such as:

$$m_1 \ddot{\delta} + d_1 \dot{\delta} + B_{12} V_1 V_2 \sin(\delta) - P_1 = 0,$$

with the residual $f_\delta$ constructed from this ODE and minimized by the network (Misyris et al., 2019). The same methodology extends to steady-state problems, parametric PDEs, and boundary value problems.

2. Training Strategies and Network Architectures

Efficiently training PINNs requires careful balancing between minimizing data loss and enforcing the physical constraints over collocation points. Key strategies include:

• Composite Loss Balancing: Weighting between data loss (matching observed/measured/boundary data) and physics loss (minimizing residuals). Empirically, a small number of training samples (as few as 40 in a power system case paper) combined with thousands of collocation points for the physics loss (e.g., 8,000) suffices to regularize the solution space and achieve high accuracy (Misyris et al., 2019).
• Network Simplicity through Physics Constraints: By incorporating physics-informed loss terms, smaller and simpler feed-forward architectures (e.g., 5 layers × 10 neurons) can outperform or match much larger networks trained with data loss alone.
• Adaptive Sampling and Domain Decomposition: For systems with strong localization, sharp gradients, or material inhomogeneities, adaptive collocation schemes and domain decomposition (e.g., partitioning the domain and imposing continuity at interfaces) can mitigate the inability of global neural ansatz functions to resolve fine-scale structure (Henkes et al., 2021).
• Parameter Identification: Unknown or uncertain system parameters (e.g., inertia $m_1$ or damping $d_1$ in electromechanical systems) may be encoded as learnable variables and identified by augmenting the PINN loss to jointly fit states and parameters, even with limited and noisy observations (Misyris et al., 2019).

3. Performance Metrics and Computational Efficiency

The practical utility of PINNs is grounded in both accuracy and computational gains relative to conventional numerical solvers:

• Accuracy: Relative $L_2$ errors of the network on physical states (e.g., rotor angle $\delta$ and frequency $\omega$) can reach the order of $1.34 \times 10^{-2}$ in challenging dynamic regimes with minimal data (Misyris et al., 2019).
• Computational Speed: Once trained, a PINN architecture can compute trajectories, dynamic states, or parametric queries in fractions of a second—e.g., $0.016$ s/trajectory for dynamic power system states, compared to $0.45$ s/trajectory for a standard ode45 numerical solver; when computing directly at a target time step, up to $87 \times$ faster than standard integration (Misyris et al., 2019).
• Parameter Recovery: System parameter estimation (e.g., inertia/damping) demonstrates prediction errors as low as $0.74\%$ and $1.28\%$, with robust identification from sparse measurements (Misyris et al., 2019).
• Scalability: The cost of PINN evaluation is independent of the state space discretization (i.e., meshless), making it promising for high-dimensional or parameterized systems.

4. Practical Relevance and Applications

PINNs offer several compelling opportunities in both forward and inverse problems:

• Realtime and Operator Inference: Networks can produce system states (e.g., rotor angles, temperatures, stress fields) at arbitrary input configurations, eliminating the need for step-by-step integration over time or space (Misyris et al., 2019).
• Inverse Problems and Parameter Identification: PINNs enable efficient data-driven recovery of system parameters and unmeasured states, supporting condition monitoring, fault diagnosis, and system identification even under uncertainty.
• System Operators and Security Assessment: Rapid, direct mapping from system input parameters (e.g., generation/load perturbations) to dynamic responses allows for faster online security assessment and control.
• State Estimation in Power Systems: The approach generalizes beyond the SMIB model, suggesting a path to high-speed, model-consistent state estimation in larger power grids, accommodating uncertainty in model parameters and system configurations.

5. Illustrative Case Study: Single-Machine Infinite Bus (SMIB) System

As an explicit example, the SMIB swing equation is written as:

$$f_\delta(t, P_1) = m_1 \ddot{\delta} + d_1 \dot{\delta} + B_{12} V_1 V_2 \sin(\delta) - P_1.$$

A PINN is trained to map $(t, P_1)$ to $\delta$, with the loss:

$$\text{MSE} = \frac{1}{N_u} \sum_{i=1}^{N_u} |\delta(t_i, P_{1,i}) - \delta_i|^2 + \frac{1}{N_x} \sum_{j=1}^{N_x} |f_\delta(t_j, P_{1,j})|^2.$$

With $N_u = 40$ data points and $N_x = 8000$ collocation points, the trained network predicts both $\delta$ and $\dot{\delta}$ accurately throughout the domain. When $m_1$ and $d_1$ are considered unknown, their values are recovered through further training.

Metric	PINN	Traditional (ode45)
Time per trajectory	0.016 s	0.45 s
Instantaneous state eval	0.004 s	0.35 s
Relative $L_2$ error ($\delta$)	$1.34 \times 10^{-2}$	N/A
Parameter estimation error	$<1.3\%$	N/A

6. Implementation Considerations and Limitations

Several factors may influence PINN performance in practice:

• Data Quality and Regularization: The method’s strong regularization via embedded physics enables training with limited and scattered data, but extreme data noise or model misspecification may degrade robustness.
• Network Architecture: Smaller networks often suffice due to the inductive bias introduced by the physics term; overparameterization may lead to overfitting when physics are weakly informative.
• Automatic Differentiation: Efficient computation of higher-order derivatives is critical, especially for stiff or highly nonlinear dynamics; frameworks such as TensorFlow enable seamless integration.
• Generalizability: While demonstrated for SMIB systems, the approach is extensible to larger power systems, hybrid models (incorporating measurements and physics), and other domains governed by differential–algebraic systems.
• Physics Model Validity: The benefit of rapid prediction and strong performance depends on the accuracy of the embedded physical law. Significant departures (e.g., unmodeled dynamics, non-standard operating regimes) may necessitate further hybridization with data-driven corrections.

7. Outlook and Future Directions

PINNs are positioned as a versatile tool for a variety of power system applications and have shown promise in accelerating system studies, reducing data requirements, and enabling parameter inference. The meshless, automatic-differentiation-based framework supports both steady-state and dynamic problems, operator learning, and can be applied to both transient and equilibrium analyses.

Areas of ongoing research include scaling to large interconnected systems, incorporating algebraic constraints and mixed physics–data regimes, developing adaptive sampling and domain decomposition strategies, and generalizing to hybrid grids with significant stochasticity and renewables. The findings in the SMIB case suggest broader applicability across physical science and engineering disciplines where differential models are known but traditional solvers are computationally burdensome or data is limited (Misyris et al., 2019).