Physics-Informed Neural Networks (PINNs)

• Physics-Informed Neural Networks (PINNs) are hybrid models that incorporate governing PDEs as soft constraints, integrating physical laws directly into the training process.
• They leverage automatic differentiation and meshless sampling to enforce accuracy and stability in simulating complex systems with limited or noisy data.
• PINNs provide a versatile framework for forward, inverse, and ill-posed problems, though careful regularization and tuning are essential to mitigate issues like local minima.

Physics-Informed Neural Network (PINN)-based Method

Physics-Informed Neural Networks (PINNs) are a class of machine learning algorithms that embed the governing physical laws of a system, typically in the form of partial differential equations (PDEs) or integral-differential equations, directly into the structure of neural network training. Rather than relying exclusively on data, PINNs incorporate the underlying physics as soft constraints in their loss functions. This hybrid approach allows accurate, robust, and data-efficient forward and inverse simulations across a variety of scientific domains, including radiative transfer and ill-posed or inverse problems.

1. Core Principles of PINN-based Algorithms

The essential idea behind PINNs is to parameterize the unknown solution $u$ of a governing differential equation via a trainable neural network $u_\theta(x)$, where $x$ denotes the set of independent variables (spatial, temporal, or otherwise) and $\theta$ are the trainable parameters (weights and biases). The network is trained by minimizing a loss function that involves both the residual of the PDE and any initial/boundary/observational data: $$\mathcal{L}(\theta) = \lambda_\mathrm{PDE} \mathcal{L}_\mathrm{PDE} + \lambda_\mathrm{data} \mathcal{L}_\mathrm{data} + \lambda_\mathrm{BC/IC} \mathcal{L}_\mathrm{BC/IC}$$ where, for example, the PDE residual loss is: $$\mathcal{L}_\mathrm{PDE} = \frac{1}{N_r} \sum_{i=1}^{N_r} \left| \mathcal{D}[u_\theta](x_i) - f(x_i) \right|^2$$ with $\mathcal{D}$ the differential operator describing the physics and $f$ the source term.

Automatic differentiation is employed to evaluate the required derivatives at collocation points, ensuring smooth and accurate enforcement of high-order or nonlocal operators.

For radiative transfer and similar applications, the loss construction is identical in philosophy—minimizing the residual of the radiative transfer equation while simultaneously enforcing consistency with available (often scarce or noisy) data and boundary/initial conditions (Mishra et al., 2020).

2. Theoretical Error Estimation and Stability

PINN-based methods are accompanied by explicit theoretical error bounds that quantitatively describe the relation between:

• The training (empirical) residual error on the discrete collocation set,
• The numerical quadrature approximation error (arising from the stochastic, meshless sampling of the domain),
• The global (generalization) error measured in the appropriate function space.

For example, error estimates derived via energy methods and Grönwall's inequality show that the $L^2$ norm of the difference between the network solution $u_\theta$ and the classical solution $u$ is bounded as

$$E_G \leq \mathrm{Const} \times \left( E_\mathrm{train} + E_\mathrm{quad} \right)$$

where the constants depend on the regularity and stability of the physical solution (e.g., norms in $C_t^0 C_x^4$ or $C_t^1 C_x^1$ space for KdV–Kawahara-type equations). This establishes that the intrinsic stability of the PDE propagates to the PINN approximation, provided the residual and boundary errors are controlled (Bai et al., 2021).

Moreover, convergence of the PINN approximation to the true solution is guaranteed when the regularizing terms (in ill-posed or inverse settings, see next section) are properly tuned and the training error converges to zero under sufficient sampling and network capacity.

3. PINN-based Methods for Inverse and Ill-posed Problems

In inverse or ill-posed problem settings, such as the Cauchy problem for elliptic equations or radiative transfer coefficient recovery, PINNs are adapted by integrating quasi-regularization (quasi-réversibilité) into the loss framework. This approach is informed by classical techniques in the theory of ill-posed problems (notably the works of R. Latt and J.-L. Lions), where a stabilizing term is introduced: $$\text{New loss:} \quad \mathcal{L}_\text{QReg}(\theta) = \mathcal{L}_\text{orig}(\theta) + \epsilon \|\mathcal{R}[u_\theta]\|_2^2$$ with $\epsilon$ a regularization parameter, and $\mathcal{R}$ a penalizing (smoothing) operator.

As $\epsilon \to 0$, the solution of the regularized problem converges, under suitable conditions, to the unique physical solution of the original ill-posed problem. In practice, this regularization:

• Stabilizes the optimization landscape by penalizing high-frequency or unstable solution modes,
• Facilitates error estimation and convergence analysis even in the presence of data that do not depend continuously on the solution,
• Implements a trade-off (via the regularization parameter) between data fidelity and numerical stability (Mishra et al., 2020).

The method is widely used to treat inverse radiative transfer, recovery of spatially varying coefficients, Cauchy problems for systems of transport/elliptic/hyperbolic type, and problems prevalent in experimental and computational mechanics.

4. Implementation and Computational Characteristics

PINN-based methods are considered straightforward to implement, especially in regimes where classical discretization-based solvers become challenging (e.g., high-dimensional, complex geometry, or coefficient-inverse problems). The training process involves:

• Selection of collocation points (using meshless strategies such as Sobol sequences or Latin hypercube sampling),
• Construction of a composite loss (equation residual, data, initial/boundary, and, for inverse problems, regularization),
• Use of established optimizers (e.g., L-BFGS for rapid convergence to local minima),
• Automatic differentiation for evaluation of all required derivatives.

Performance metrics such as empirical training error, generalization error (via cross-validation on analytic solutions or fine-grid reference results), and computational time (a few seconds to minutes, depending on problem complexity and network size) are commonly reported. For nonlinear dispersive PDEs with soliton solutions, PINNs achieve relative errors well below 0.1% and significantly reduced computational overhead compared to traditional solvers (Bai et al., 2021).

5. Comparative Advantages and Limitations

PINNs exhibit several advantages:

• Ease of Implementation and Meshlessness: No need for mesh generation or bespoke solvers; the only requirement is encoding the PDE and loss in an autodiff-capable framework.
• Versatility: Applicable to forward, inverse, and ill-posed problems in the same unified architecture.
• Stability and Robustness: Error estimates rooted in the underlying PDE stability; robust even with limited or noisy data, especially when regularization is enforced.
• Efficiency: For high-dimensional or data-scarce regimes, PINNs may outperform classical methods in both speed and scalability.

However, certain limitations are observed:

• For highly oscillatory or multiscale problems, direct PINN application may fail without preconditioning, adaptive sampling, or hybridization with classical homogenization techniques.
• Training can converge to local minima, especially in non-convex (ill-posed/inverse) settings. Careful tuning of regularization is required to guarantee convergence and control overfitting.
• For some classes of stiff PDEs or near-singular solutions, network architecture and optimizer selection significantly impact final accuracy.

6. Outlook, Applications, and Context

PINN-based methodologies are broadly applicable in computational physics, engineering, and applied mathematics:

• Radiative transfer simulations with forward and inverse objectives,
• Computational mechanics, particularly for parameter identification and assimilation of partial data,
• Inverse problems, including Cauchy problems and parameter recovery in ill-posed scenarios,
• Experiment-driven modeling where direct measurement of all state variables is not feasible.

The combination of quasi-regularization theory with modern neural optimization, as exemplified in applications to radiative transfer and related fields, provides a foundation for further research into hybrid, adaptive, and multi-fidelity methods. Through integration of regularization, rigorous error estimates, and meshless frameworks, PINNs are positioned as efficient and practical alternatives to both traditional numerical solvers and classical regularization-based inverse methods in a variety of scientific and engineering contexts (Mishra et al., 2020).