Physics-Informed Neural Networks: Fundamental Limitations and Conditional Usefulness

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Physics-Informed Neural Networks (PINNs) aim to approximate the solution \( u \) of a differential equation defined over spatial coordinates \( x \in \mathbb{R}^d \) (e.g., \( x = (x_1, x_2, x_3) \)), and, when applicable,  time \( t \), by representing \( u \) with a neural network \( u_\theta \), where \( \theta \) denotes the trainable weights and biases. Training involves minimizing a composite loss function  $\mathcal{L}(\theta) = \mathcal{L}_{\text{PDE}} + \mathcal{L}_{\text{BC}} + \mathcal{L}_{\text{IC}} + \mathcal{L}_{\text{data}},$ which enforces the governing PDE, boundary conditions, initial conditions, and any available observational data.  

However, PINNs minimize the PDE residual only indirectly—by adjusting the neural network parameters rather than manipulating solution components or their derivatives in a controlled, explicit manner. This leads to several fundamental inefficiencies. Since the solution is represented by a neural network, the required derivatives arise from symbolic differentiation of its functional form, not from discretized numerical schemes such as finite differences or finite elements. Consequently, the behavior of these derivatives is governed entirely by the shape of the learned function, without the stabilizing structure provided by meshes or derivative stencils. This lack of control makes it difficult to maintain a consistent balance between different derivative terms during training. In stiff or high-order PDEs, where multiple scales or high-order derivatives interact, the resulting optimization problem often suffers from poor gradient flow, instability, or prohibitively slow convergence.

Unlike classical numerical methods, such as finite difference or finite element approaches, which embed structural information—like conservation laws or derivative couplings—directly into the computational framework, PINNs must infer such structure from the loss landscape alone. This can result in solutions that minimize residuals locally while failing to capture the true global behavior of the PDE, particularly in regions where errors cancel or gradients vanish. Additionally, since PINNs encode geometry only implicitly through collocation point sampling, and lack explicit mesh or parametric domain representations, they are poorly suited for domains with complex, irregular, or evolving geometry.

Despite these limitations, PINNs may still offer practical value in certain contexts. Specifically, they are advantageous when the solution is influenced by hidden dynamics, uncertain initial conditions, or prior states not fully encoded in the PDE itself. A representative example is option pricing in finance: while the Black--Scholes equation assumes constant volatility, real markets exhibit time-varying behavior influenced by historical trends. In such cases, PINNs can incorporate historical price data into the training process and learn corrections to the idealized PDE, enabling more realistic and flexible modeling.

 

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