Physics-Informed Neural Networks (PINNs) for Parameter Estimation in Nonlinear Differential Equations
Kaludura, Chamath Piyum Suraj De Silva (2025)
2025
Master's Programme in Computing Sciences and Electrical Engineering
Informaatioteknologian ja viestinnän tiedekunta - Faculty of Information Technology and Communication Sciences
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Hyväksymispäivämäärä
2025-05-19
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:tuni-202505195728
https://urn.fi/URN:NBN:fi:tuni-202505195728
Tiivistelmä
This thesis explores the application of physics-informed neural networks (PINNs) for parameter estimation in nonlinear dynamical systems, with a focus on the damped harmonic oscillator. Traditional numerical approaches for solving such systems often face challenges related to computational cost, sensitivity to noise, and dependence on large amounts of clean data. PINNs provide a modern alternative by embedding the governing physical laws directly into the training process of neural networks, resulting in models that are both data-effcient and physically consistent.
We implemented inverse PINNs using PyTorch-based feedforward neural networks with Tanh activation. The model was trained using the Adam optimizer on a hybrid loss function that combines data fdelity with physics-based residuals weighted by a fxed coeffcient. Parameters such as damping and stiffness were treated as trainable variables, enabling the network to learn them directly from observed trajectories. Training and validation were performed using synthetically generated data from numerically solved differential equations.
Experimental results demonstrated that inverse PINNs accurately estimated physical parameters with Relative errors below 2%, maintained low root mean squared error (RMSE ≈ 0.00285) in both in-range predictions and out-of-range (future prediction) tasks, and remained robust under moderate observational noise. The standard errors (SE) for both µ and k dropped sharply and stabilized after 10 samples, indicating a high degree of confdence in the parameter estimates. In contrast, the forward model required over 50 samples to achieve similar SE stabilization, showing higher sample dependency. These results confrm that inverse PINNs exhibit fast stabilization and strong data effciency, while forward PINNs also reach accurate solutions given suffcient data. The fndings highlight the impact of noise on physics-informed learning and the importance of evaluating model robustness under different noise levels.
Overall, this work establishes inverse PINNs as a robust and scalable approach for parameter estimation and system identifcation in physics-based models, particularly where data is limited or noisy. Future directions include incorporating uncertainty quantifcation, adaptive loss balancing, and expanding applications to high-dimensional and real-world systems.
This work applies inverse Physics-Informed Neural Networks (PINNs) to estimate parameters in nonlinear dynamical systems, using a PyTorch-based model trained on a hybrid loss function that combines data accuracy with physical consistency. Experimental evaluations on damped harmonic oscillators demonstrated high parameter estimation accuracy, strong generalization to unseen time domains through out-of-range predictions, and robustness to noise, establishing inverse PINNs as an effective alternative to traditional methods like the Multiple Shooting Method.
We implemented inverse PINNs using PyTorch-based feedforward neural networks with Tanh activation. The model was trained using the Adam optimizer on a hybrid loss function that combines data fdelity with physics-based residuals weighted by a fxed coeffcient. Parameters such as damping and stiffness were treated as trainable variables, enabling the network to learn them directly from observed trajectories. Training and validation were performed using synthetically generated data from numerically solved differential equations.
Experimental results demonstrated that inverse PINNs accurately estimated physical parameters with Relative errors below 2%, maintained low root mean squared error (RMSE ≈ 0.00285) in both in-range predictions and out-of-range (future prediction) tasks, and remained robust under moderate observational noise. The standard errors (SE) for both µ and k dropped sharply and stabilized after 10 samples, indicating a high degree of confdence in the parameter estimates. In contrast, the forward model required over 50 samples to achieve similar SE stabilization, showing higher sample dependency. These results confrm that inverse PINNs exhibit fast stabilization and strong data effciency, while forward PINNs also reach accurate solutions given suffcient data. The fndings highlight the impact of noise on physics-informed learning and the importance of evaluating model robustness under different noise levels.
Overall, this work establishes inverse PINNs as a robust and scalable approach for parameter estimation and system identifcation in physics-based models, particularly where data is limited or noisy. Future directions include incorporating uncertainty quantifcation, adaptive loss balancing, and expanding applications to high-dimensional and real-world systems.
This work applies inverse Physics-Informed Neural Networks (PINNs) to estimate parameters in nonlinear dynamical systems, using a PyTorch-based model trained on a hybrid loss function that combines data accuracy with physical consistency. Experimental evaluations on damped harmonic oscillators demonstrated high parameter estimation accuracy, strong generalization to unseen time domains through out-of-range predictions, and robustness to noise, establishing inverse PINNs as an effective alternative to traditional methods like the Multiple Shooting Method.