Beyond Residuals: Energy based solutions of partial differential equations using scientific machine learning
Abstract
Energy-based approaches provide a natural and physically consistent framework for a large class of partial differential equations arising in solid and fluid mechanics, where the governing equations follow from variational principles. In contrast to residual-based physics-informed neural networks (PINNs) and their weak-form variants, which enforce the strong or weak form of the equations through loss minimization, the Deep Energy Method (DEM) directly computes the solution as the minimizer of an energy or incremental potential functional. This eliminates the need for residual weighting, avoids high-order derivatives, and enables the direct enforcement of physical constraints through the variational structure.In this work, we systematically revisit the Deep Energy Method, placing it in the broader context of physics-informed learning and variational modeling. We clarify the relationship between DEM, PINNs, and VPINNs, and identify the class of problems for which energy minimization provides intrinsic advantages in terms of stability, robustness and interpretability. Particular emphasis is placed on incremental variational formulations, which allow DEM to be applied to nonlinear, history-dependent and time-dependent problems, including phase-field fracture and dissipative systems. The variational structure underlying DEM further provides a natural foundation for optimization and inverse problems, where the energy functional acts as a physics-based constraint rather than a residual penalty. Through a series of numerical examples, we demonstrate that DEM offers a principled and effective alternative to residual-based methods for variational problems, highlighting its strengths and limitations relative to existing physics-informed approaches.
Related
- A Generalized Template Framework for Constructing Exact Analytical Solutions to Various Nonlinear Partial Differential Equations
- Hybrid Analytical Numerical and Machine Learning Frameworks for Solving Deterministic and Stochastic Differential Equations with Stability, Convergence, and Uncertainty Quantification
- How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes
- Bayesian data-driven discovery of partial differential equations with variable coefficients
- Fast quantum algorithm for differential equations
BibTeX
@misc{rabczuk2026beyond,
title = {{Beyond Residuals: Energy based solutions of partial differential equations using scientific machine learning}},
author = {Timon Rabczuk and Yizheng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08239},
eprint = {2609.08239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08239},
}
RIS
TY - UNPB
TI - Beyond Residuals: Energy based solutions of partial differential equations using scientific machine learning
AU - Timon Rabczuk
AU - Yizheng Wang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08239
UR - https://arxiv.org/abs/2609.08239
AB - Energy-based approaches provide a natural and physically consistent framework for a large class of partial differential equations arising in solid and fluid mechanics, where the governing equations follow from variational principles. In contrast to residual-based physics-informed neural networks (PINNs) and their weak-form variants, which enforce the strong or weak form of the equations through loss minimization, the Deep Energy Method (DEM) directly computes the solution as the minimizer of an energy or incremental potential functional. This eliminates the need for residual weighting, avoids high-order derivatives, and enables the direct enforcement of physical constraints through the variational structure.In this work, we systematically revisit the Deep Energy Method, placing it in the broader context of physics-informed learning and variational modeling. We clarify the relationship between DEM, PINNs, and VPINNs, and identify the class of problems for which energy minimization provides intrinsic advantages in terms of stability, robustness and interpretability. Particular emphasis is placed on incremental variational formulations, which allow DEM to be applied to nonlinear, history-dependent and time-dependent problems, including phase-field fracture and dissipative systems. The variational structure underlying DEM further provides a natural foundation for optimization and inverse problems, where the energy functional acts as a physics-based constraint rather than a residual penalty. Through a series of numerical examples, we demonstrate that DEM offers a principled and effective alternative to residual-based methods for variational problems, highlighting its strengths and limitations relative to existing physics-informed approaches.
ER -