# PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks

arXiv: https://arxiv.org/abs/2504.05248 · HTML: https://mlsanity.com/paper/2504.05248/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 79 references that are in the map.
- First posted on arXiv 7 Apr 2025.

## Builds on (most cited first, 27 of 27)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Sheared LLaMA: Accelerating Language Model Pre-training via Structured Pruning](https://mlsanity.com/paper/2310.06694/) (2023)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models](https://mlsanity.com/paper/2208.06677/) (2022)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [Controlled Text Generation as Continuous Optimization with Multiple Constraints](https://mlsanity.com/paper/2108.01850/) (2021)
- [DiSECt: A Differentiable Simulation Engine for Autonomous Robotic Cutting](https://mlsanity.com/paper/2105.12244/) (2021)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networks](https://mlsanity.com/paper/2105.00862/) (2021)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [Training neural networks under physical constraints using a stochastic augmented Lagrangian approach](https://mlsanity.com/paper/2009.07330/) (2020)
- [Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE](https://mlsanity.com/paper/2012.06691/) (2020)
- [NSGA-PINN: A Multi-Objective Optimization Method for Physics-Informed Neural Network Training](https://mlsanity.com/paper/2303.02219/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [PINN Training using Biobjective Optimization: The Trade-off between Data Loss and Residual Loss](https://mlsanity.com/paper/2302.01810/) (2023)
- [Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.08313/) (2024)
- [Bayesian Entropy Neural Networks for Physics-Aware Prediction](https://mlsanity.com/paper/2407.01015/) (2024)
- [Neural Conjugate Flows: Physics-informed architectures with flow structure](https://mlsanity.com/paper/2411.08326/) (2024)

## Built on by (most cited first, 4 of 4)

- [LLM-Based Test Oracles: Source-of-Authority Taxonomy -- A Systematic Literature Review](https://mlsanity.com/paper/2607.05031/) (2026)
- [Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning](https://mlsanity.com/paper/2505.11682/) (2025)
- [On the Nonasymptotic Scaling Guarantee of Hyperparameter Estimation in Inhomogeneous, Weakly-Dependent Complex Network Dynamical Systems](https://mlsanity.com/paper/2601.15603/) (2026)
- [Hyperparameter selection for equation learning with biologically-informed neural networks](https://mlsanity.com/paper/2610.02954/) (2026)

## Abstract (arXiv)

Parameter estimation for differential equations from measured data is an inverse problem prevalent across quantitative sciences. Physics-Informed Neural Networks (PINNs) have emerged as effective tools for solving such problems, especially with sparse measurements and incomplete system information. However, PINNs face convergence issues, stability problems, overfitting, and complex loss function design. Here we introduce PINNverse, a training paradigm that addresses these limitations by reformulating the learning process as a constrained differential optimization problem. This approach achieves a dynamic balance between data loss and differential equation residual loss during training while preventing overfitting. PINNverse combines the advantages of PINNs with the Modified Differential Method of Multipliers to enable convergence on any point on the Pareto front. We demonstrate robust and accurate parameter estimation from noisy data in four classical ODE and PDE models from physics and biology. Our method enables accurate parameter inference also when the forward problem is expensive to solve.