# DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation

arXiv: https://arxiv.org/abs/2012.02681 · HTML: https://mlsanity.com/paper/2012.02681/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2020.

## Built on by (most cited first, 30 of 45)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [Dual Cone Gradient Descent for Training Physics-Informed Neural Networks](https://mlsanity.com/paper/2409.18426/) (2024)
- [Recurrent Neural Operators: Stable Long-Term PDE Prediction](https://mlsanity.com/paper/2505.20721/) (2025)
- [SPATE-GAN: Improved Generative Modeling of Dynamic Spatio-Temporal Patterns with an Autoregressive Embedding Loss](https://mlsanity.com/paper/2109.15044/) (2021)
- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
- [Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2205.14630/) (2022)
- [Efficient Error Certification for Physics-Informed Neural Networks](https://mlsanity.com/paper/2305.10157/) (2023)
- [Understanding and Mitigating Extrapolation Failures in Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.09478/) (2023)
- [Neural oscillators for generalization of physics-informed machine learning](https://mlsanity.com/paper/2308.08989/) (2023)
- [Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulations](https://mlsanity.com/paper/2311.00578/) (2023)
- [TimeKit: A Time-series Forecasting-based Upgrade Kit for Collaborative Filtering](https://mlsanity.com/paper/2211.04266/) (2022)
- [Personalized Predictions of Glioblastoma Infiltration: Mathematical Models, Physics-Informed Neural Networks and Multimodal Scans](https://mlsanity.com/paper/2311.16536/) (2023)
- [ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms](https://mlsanity.com/paper/2512.03476/) (2025)
- [Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations](https://mlsanity.com/paper/2206.09571/) (2022)
- [Frequency-adaptive Multi-scale Deep Neural Networks](https://mlsanity.com/paper/2410.00053/) (2024)
- [Dual-Balancing for Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.11117/) (2025)
- [Network-wide Freeway Traffic Estimation Using Sparse Sensor Data: A Dirichlet Graph Auto-Encoder Approach](https://mlsanity.com/paper/2503.15845/) (2025)
- [On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition](https://mlsanity.com/paper/2507.08861/) (2025)
- [GRINN: A Physics-Informed Neural Network for solving hydrodynamic systems in the presence of self-gravity](https://mlsanity.com/paper/2308.08010/) (2023)
- [Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN)](https://mlsanity.com/paper/2401.02300/) (2024)
- [A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments](https://mlsanity.com/paper/2507.10792/) (2025)
- [Neuro-Spectral Architectures for Causal Physics-Informed Networks](https://mlsanity.com/paper/2509.04966/) (2025)
- [Physics-Informed Graph Learning](https://mlsanity.com/paper/2202.10679/) (2022)
- [Modelling of physical systems with a Hopf bifurcation using mechanistic models and machine learning](https://mlsanity.com/paper/2209.06910/) (2022)
- [MoGERNN: An Inductive Traffic Predictor for Unobserved Locations](https://mlsanity.com/paper/2501.12281/) (2025)

## Abstract (arXiv)

We present a method for learning dynamics of complex physical processes described by time-dependent nonlinear partial differential equations (PDEs). Our particular interest lies in extrapolating solutions in time beyond the range of temporal domain used in training. Our choice for a baseline method is physics-informed neural network (PINN) [Raissi et al., J. Comput. Phys., 378:686--707, 2019] because the method parameterizes not only the solutions but also the equations that describe the dynamics of physical processes. We demonstrate that PINN performs poorly on extrapolation tasks in many benchmark problems. To address this, we propose a novel method for better training PINN and demonstrate that our newly enhanced PINNs can accurately extrapolate solutions in time. Our method shows up to 72% smaller errors than existing methods in terms of the standard L2-norm metric.