New Method Provides Rigorous Error Bounds for Physics-Informed Neural Networks
A new preprint introduces computable lower and upper a posteriori error bounds for physics-informed neural networks (PINNs) applied to ordinary differential equations. Prior work had established upper bounds, but the absence of complementary lower bounds left PINN predictions without full error certification. The framework enables rigorous, two-sided error enclosures without requiring access to the exact solution, potentially improving the reliability of PINNs in scientific and engineering applications.
Researchers have proposed a framework for certifying the accuracy of physics-informed neural networks (PINNs) when solving ordinary differential equations (ODEs), addressing a longstanding gap in the field. While previous work had produced rigorous upper bounds on PINN prediction errors, no computable lower bounds existed, meaning practitioners could not fully bracket the true error. The new approach derives lower bounds under a localized strong monotonicity condition and pairs them with upper bounds under a one-sided Lipschitz condition—a weaker assumption than the global Lipschitz condition used in earlier studies, which can yield tighter upper error bands. For linear time-invariant and time-varying systems, the authors provide explicit formulas expressed in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. The paper also examines how the method of enforcing initial conditions—soft versus hard—affects the informativeness of the lower certificate, and proposes a signed-residual finite-probe approach to recover nontrivial lower bounds in the linear setting. Additionally, a certificate-informed training strategy is introduced, using the upper certificate as an auxiliary regularizer during training while reserving lower certificates as post-training diagnostics. The work is presented as a preprint on arXiv and has not yet undergone formal peer review.
What's missing
The study is a preprint and has not yet been peer-reviewed. Key open questions include: how the framework scales to partial differential equations (PDEs) and higher-dimensional systems; whether the localized strong monotonicity and one-sided Lipschitz conditions are readily verifiable in practice for complex real-world problems; and how the tightness of the bounds compares empirically across a broad range of ODE types and PINN architectures.
What different sources said
- arXiv cs.LGCenter
Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds
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