Title: PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: physics informed neural networks, parameter estimation, inverse problems, machine learning

Abstract: Estimating unknown parameters in differential equations from noisy, sparse data is a common inverse problem in science and engineering. While Physics-Informed Neural Networks (PINNs) have shown promise, their standard training paradigm, which relies on a weighted-sum loss, often leads to overfitting and fails to enforce physical laws in the presence of noise. This failure stems from the inability of gradient-based methods to find balanced solutions on the complex, often non-convex, Pareto fronts that arise in such multi-objective settings.
We introduce PINNverse, a new training paradigm that overcomes these limitations by reformulating the learning process as a constrained optimization problem. Instead of balancing competing objectives with ad-hoc weights, PINNverse minimizes the data-fitting error subject to the explicit constraint that the differential equations and boundary conditions are satisfied. To solve this, we employ the Modified Differential Method of Multipliers (MDMM). By simultaneously updating network weights and Lagrange multipliers (via gradient ascent) in a single optimization loop, this method avoids the expensive nested loops required by conventional augmented Lagrangian techniques and seamlessly integrates with standard optimizers like Adam. This enables convergence to any point on the Pareto front---including concave regions inaccessible to standard PINNs---while adding negligible computational overhead.
Experiments on four challenging ODE and PDE benchmarks demonstrate that PINNverse achieves robust and accurate parameter estimation even with significant data noise and poor initial guesses, successfully preventing overfitting and ensuring strict adherence to the governing physics. By solving the forward and inverse problems concurrently, PINNverse enables efficient parameter inference in systems where repeated forward evaluations with classical numerical solvers would be computationally prohibitive.

TL;DR: A new PINN training method that stops them from overfitting to noisy data, letting them find the correct physical parameters for almost no extra compute cost