On the Role of Consistency Between Physics and Data in Physics-Informed Neural Networks

TL;DR

This work identifies a consistency barrier in physics-informed neural networks (PINNs): when training data do not perfectly satisfy the PDE, the attainable accuracy is intrinsically limited even with exact enforcement of the PDE residual. Using a controlled 1D viscous Burgers equation with a manufactured solution, the authors quantify how data fidelity governs PINN convergence and accuracy, comparing fixed-weight and adaptive loss weighting (lbPINN) across several data-consistency scenarios. The study shows that high-fidelity numerical data or analytical data can effectively remove the barrier, while low-to-medium fidelity data induce saturation near the data’s discretization error; the barrier also manifests in Pareto-front analyses of PDE versus data losses. These findings emphasize the critical role of data quality and loss balancing in practical PINN applications and provide guidance for constructing reliable physics-informed surrogates, with extensions to higher dimensions and uncertainty-aware formulations as future directions.

Abstract

Physics-informed neural networks (PINNs) have gained significant attention as a surrogate modeling strategy for partial differential equations (PDEs), particularly in regimes where labeled data are scarce and physical constraints can be leveraged to regularize the learning process. In practice, however, PINNs are frequently trained using experimental or numerical data that are not fully consistent with the governing equations due to measurement noise, discretization errors, or modeling assumptions. The implications of such data-to-PDE inconsistencies on the accuracy and convergence of PINNs remain insufficiently understood. In this work, we systematically analyze how data inconsistency fundamentally limits the attainable accuracy of PINNs. We introduce the concept of a consistency barrier, defined as an intrinsic lower bound on the error that arises from mismatches between the fidelity of the data and the exact enforcement of the PDE residual. To isolate and quantify this effect, we consider the 1D viscous Burgers equation with a manufactured analytical solution, which enables full control over data fidelity and residual errors. PINNs are trained using datasets of progressively increasing numerical accuracy, as well as perfectly consistent analytical data. Results show that while the inclusion of the PDE residual allows PINNs to partially mitigate low-fidelity data and recover the dominant physical structure, the training process ultimately saturates at an error level dictated by the data inconsistency. When high-fidelity numerical data are employed, PINN solutions become indistinguishable from those trained on analytical data, indicating that the consistency barrier is effectively removed. These findings clarify the interplay between data quality and physics enforcement in PINNs providing practical guidance for the construction and interpretation of physics-informed surrogate models.

Figures (7)

• Figure 1: Distribution of all dataset points used in training and test. Top: Standard PINN. 1000 uniformed distributed collocation points, 4 Train and Test data points, and 1 BC points. Bottom: Parametric PINN, 1500 Sobol distributed collocation points, $3\times 81$ Train and Test data points, and 50 uniformly distributed BC points.
• Figure 2: Interpreting the optimization of the standard PINN via minimization of Eq. \ref{['eq:pareto_loss']} as a (scalarized) MOO problem, we show, in the $(\mathcal{L}_\text{PDE}, \mathcal{L}_\text{D})$ plane, all feasible solutions achieved throughout optimization, for each value of $\alpha$ (color coded). The final values of each optimization (highlighted in white dots) denote Pareto-optimal solutions for each trade-off weight $\alpha$, and altogether approximate the Pareto front. Each panel depicts this information obtained for the four consistency scenarios (C1-C3 and Analytical). Additionally, each panel also depicts the optimization trajectory of the standard PINN obtained by minimizing the Eq. \ref{['eq:dynamic_loss']} given by the lbPINN approach (solid lines). The final solution of this optimization is indicated by a star-shaped marker in the corresponding color. Overall, lbPINN attains solutions that are compatible with the estimated Pareto front of the fixed-weight loss Eq. \ref{['eq:pareto_loss']}.
• Figure 3: testRMSE (Eq. \ref{['eq:testRMSE']}, measuring the error between the PINN's prediction and the numerical data solution) as a function of the number of training iterations, for the standard PINN case trained on the four consistency scenarios. The switch between an AdamW and a L-BFGS optimizer is highlighted.
• Figure 4: Spatially-dependent absolute error $|u_\theta(x)-u(x)|$ (solid lines) between the PINN prediction and the analytical solution, for the standard PINN trained on the four different consistency scenarios. For comparison, in dashed lines the spatially-dependent absolute error of the numerical data $|\tilde{u}(x)-u(x)|$ is also depicted.
• Figure 5: testRMSE (Eq. \ref{['eq:testRMSE']}) as a function of the number of training iterations for the parametric PINN in all four consistency scenarios.