Critical evaluation of PINN for FWD inverse analysis and differentiable FEM as an alternative

Technical Summary: Critical Evaluation of PINN for FWD Inverse Analysis and Differentiable FEM as an Alternative

Problem Statement
The Falling Weight Deflectometer (FWD) is a standard device for non-destructive structural evaluation of pavement systems. The core challenge in FWD analysis is the "backcalculation" inverse problem: determining the elastic moduli of pavement layers from measured surface deflection basins. Traditional gradient-based optimization methods rely on finite-difference (FD) approximations for gradients. This approach suffers from poor scalability (requiring $N+1$ forward simulations for $N$ unknown moduli) and numerical instability due to truncation errors and sensitivity to perturbation step sizes. While Automatic Differentiation (AD) offers a principled alternative for exact gradient computation, its application to FWD backcalculation via Physics-Informed Neural Networks (PINNs) remains unexplored. PINNs face specific challenges in this context, including spectral bias (difficulty learning sharp discontinuities at layer interfaces) and sensitivity to hyperparameter tuning.

Methodology
The study evaluates two AD-based inverse analysis frameworks on a synthetic benchmark of a three-layer axisymmetric pavement system (surface course, base, and subgrade):

1. Differentiable Finite Element Method (DiffFEM): This approach embeds a standard linear elastic FEM forward solver within an AD framework (PyTorch). The governing physics are enforced as hard constraints by solving the linear system $K(E)u = F$ exactly at every iteration. The inverse problem optimizes only the layer moduli $E$, with gradients computed via reverse-mode AD.
2. Physics-Informed Neural Networks (PINNs):
   • Vanilla PINN: A single neural network approximates the displacement field. Physics are enforced as soft constraints via a loss function combining data misfit and physics residuals.
   • Extended PINN with Domain Decomposition (XPINN): To address the discontinuities in layered systems, the domain is decomposed, with each layer assigned a dedicated sub-network. Interface continuity is enforced via a soft penalty term in the loss function.

Both methods use the L-BFGS optimizer. The benchmark tests noise-free conditions and various levels of Gaussian measurement noise ($\sigma_n \in \{0, 1, 2, 3, 4, 5\}$ $\mu$m).

Key Contributions

• Critical Assessment of PINNs for Layered Systems: The paper demonstrates that standard PINNs fail to recover layer moduli in multilayered pavement systems due to sharp modulus discontinuities.
• Evaluation of XPINN: It investigates XPINN as a remedy for discontinuities, revealing that while it converges in noise-free cases, it remains highly sensitive to loss weighting and network architecture.
• Comparison with DiffFEM: It establishes DiffFEM as a robust alternative that enforces physics natively, offering a direct comparison between "soft" physics enforcement (PINN) and "hard" physics enforcement (DiffFEM) in an AD context.

Results

• Noise-Free Inversion:
  • Vanilla PINN: Failed to recover moduli, converging to arbitrary values due to spectral bias.
  • XPINN: Converged after $\approx 4 \times 10^5$ iterations but yielded relative errors of up to 6.9% for the subgrade. Performance was highly sensitive to the ratio of data-to-physics loss weights ($\lambda_{data}/\lambda_{phy}$) and network architecture.
  • DiffFEM: Converged in $\approx 40$ iterations with relative errors on the order of $O(10^{-9})$, recovering moduli to machine precision.
• Noisy Inversion:
  • XPINN: Performance degraded significantly under noise. At $\sigma_n = 1$ $\mu$m, it failed to identify correct moduli (errors >70%), finding a displacement field that fit sensor data but violated global equilibrium. At $\sigma_n = 5$ $\mu$m, errors became erratic and severe (base modulus error up to 391%).
  • DiffFEM: Demonstrated robust, graceful degradation. Even at $\sigma_n = 5$ $\mu$m, modulus errors remained bounded (max 14% for surface, 6.2% for base, 0.11% for subgrade).
• Computational Efficiency: DiffFEM completed optimization in approximately 60 seconds, whereas XPINN required roughly 10 hours—over two orders of magnitude slower.

Significance and Claims
The paper claims that for inverse problems where the forward solver is well-established, numerically stable, and differentiable (such as quasi-static layered elastic systems), DiffFEM is a superior choice to PINNs. DiffFEM offers higher accuracy, robustness against measurement noise, and significantly greater computational efficiency because it treats governing equations as hard constraints rather than soft penalties.

The authors argue that PINNs should be used with caution in this specific context, as they struggle with sharp material discontinuities and require extensive, problem-specific hyperparameter tuning. However, the paper acknowledges that PINNs remain attractive when differentiable forward solvers are unavailable, difficult to implement (e.g., problems with non-smooth constraints or adaptive time-stepping), or when mesh-free formulations are required. The study suggests that the choice between PINN and DiffFEM depends critically on the availability of an efficient, differentiable forward solver.