Energy-Based Formulation

• Energy-Based Formulation is a mathematical framework that leverages energy functionals to derive governing equations, ensuring conservation and controlled dissipation.
• It is applied across domains like numerical PDEs, multiphysics simulations, and machine learning, where structure-preserving discretizations and variational methods are key.
• The framework enables modular model assembly, stability through energy minimization, and optimized parameter selection via constrained minimization techniques.

An energy-based formulation is a mathematical, variational, or algorithmic framework in which the energy (or an energy-like functional) is central to the model’s structure, its governing equations, its coupling schemes, or its optimization objectives. Such formulations are fundamental across disciplines: from statistical mechanics and multiphysics modeling, through numerical methods for PDEs, to machine learning. The common thread is that energy (and its generalizations—free energy, exergy, Hamiltonians) determines both the dynamics and the admissibility of solutions, enforces conservation, encodes dissipation, or defines loss functions with physical grounding.

1. Fundamental Structure and Principles

Energy-based formulations arise by specifying an energy functional $E$ (Hamiltonian, Helmholtz free energy, potential energy, etc.)—either over continuous fields, discrete variables, or functionals of probability distributions. The governing equations are then derived as Euler–Lagrange equations, by constrained minimization, or as dissipative gradient flows. This approach ensures intrinsic compatibility with conservation laws, passivity, and thermodynamically meaningful dissipation.

Classical kinetic and fluid systems: The energy-based volumetric lattice Boltzmann method defines energy at the discrete level, ensuring exact conservation of density, momentum, and energy via a specifically crafted collision operator and Hermite-expansion of equilibria (Sbragaglia et al., 2010). In continuum and finite-element frameworks, energy-minimization—subject to constraints—yields optimal basis functions and ensures robustness across heterogeneous coefficients (e.g., in the CEM-GMsFEM for parabolic equations (Wang et al., 2020)).

Port-Hamiltonian and generalized structures: These frameworks define an energy function $H$ and use structure/dissipation matrices $J, R$, ensuring that evolution equations respect the power-balance equation

$$\dot H = \langle \text{effort variables}, \text{flow variables} \rangle - \text{dissipation}$$

and remain closed under interconnection (Altmann et al., 18 Jun 2024, Altmann et al., 17 Apr 2025).

Thermodynamics: Energy–entropy flows and zero entropy generation underpin the ideal efficiency analysis of light-powered systems, where the energy-based flow formalism delineates all loss channels and efficiency bounds (Yabuki, 28 Jan 2025). In macroscopic electrodynamics, only the vacuum tensor constructed from $\mathbf{E}, \mathbf{B}$ leads to a physically consistent energy–momentum balance (Westhoff, 12 Apr 2025).

2. Mathematical and Variational Frameworks

General mathematical form:

• State variables: $z = (z_1, z_2, z_3)$ (“effort”, “flow”, and “algebraic” components)
• Stored energy: $H(z_1, z_2)$
• DAE/port-Hamiltonian equations:

$$\begin{pmatrix} \partial_{z_1}H \ \dot{z}_2 \ 0 \end{pmatrix} = (J - R) \begin{pmatrix} \dot{z}_1 \ \partial_{z_2}H \ z_3 \end{pmatrix} + Bu$$

with skew-symmetric $J$, positive-semidefinite $R$, and input map $B$ (Altmann et al., 18 Jun 2024, Altmann et al., 17 Apr 2025).

Variational discretizations: Several energy-based formulations are built on finite-volume, finite-element, or spectral Petrov–Galerkin discretizations, preserving energy conservation or dissipation at the discrete level. For instance, discrete midpoint and discrete-gradient schemes guarantee that the dissipation property of the continuous system holds exactly in time-integration (Altmann et al., 18 Jun 2024).

Constrained minimization: In the CEM-GMsFEM setting, velocity multiscale basis functions are found via constrained minimization of a local energy subject to mass conservation, ensuring optimal support and robustness to coefficient contrast (Wang et al., 2020). Similar variational structure underpins mixed finite element elastodynamics, lattice Boltzmann solvers, and energy-function-based machine learning losses.

3. Applications in Physical and Engineering Sciences

Thermofluid-Multiphysics and PDEs

• Lattice Boltzmann and kinetic methods: The energy-based TVLBM allows flexible velocity sets, enforces energy conservation at the discrete level, and supports general grids without strict “streaming” requirements. Boundary conditions and grid refinement are naturally compatible with energy conservation (Sbragaglia et al., 2010).
• Compositional multiphysics: The Exergetic Port-Hamiltonian framework (EPHS) provides a modular syntax for coupling fluid, thermal, electrical, and electromagnetic subsystems while automatically preserving first and second laws (energy and entropy production), including Onsager symmetry (Lohmayer et al., 13 Sep 2024).
• Atomistic/continuum coupling: The energy-based blended quasicontinuum (BQCE) method interpolates between atomistic and continuum energies via a smooth blending function, balancing continuum coarsening error with interface "ghost forces," and delivering provable convergence even in the presence of crystal defects (Luskin et al., 2011).
• Electromagnetic field/circuit coupled systems: Generalized energy-based models robustly couple PDE-based field models (including eddy current losses or hysteresis) with circuit descriptions (MNA), preserving passivity and correct energy and dissipation accounting (Altmann et al., 17 Apr 2025, Egger et al., 19 Jul 2025).

Thermodynamic, Statistical, and Efficiency Analyses

• Light-powered/thermal systems: The unified energy–entropy flow approach yields compact efficiency bounds for open and closed photon engines, correctly classifying decades-old limits (Jeter, Spanner, Petela, Landsberg) as special cases of a master formula relating inflow energy and entropy to allowed discard channels (Yabuki, 28 Jan 2025).
• Macroscopic electrodynamics: Only energy-based formulations built on the vacuum Maxwell energy–momentum tensor are compatible with fundamental force–energy consistency; attempts to attribute physical reality to $\mathbf{D}, \mathbf{H}$ or mixed-tensor expressions lead to paradoxes or indeterminacy in force distribution (Westhoff, 12 Apr 2025).

Data Science and Machine Learning

• Energy-based generative modeling: In EBGAN, the discriminator defines an unnormalized energy surface, assigning low energy near the data manifold and shaping generator learning through an energy minimum game. Reconstruction-based energy discriminators offer stable gradients and support high-dimensional, high-resolution outputs (Zhao et al., 2016).
• Physically-consistent losses: Loss functions explicitly derived from Boltzmann distributions around data points (energy-based losses) promote physically meaningful learning, symmetry-respect, and improved performance in molecular generation and spin systems (Kaba et al., 3 Nov 2025).

Optimization and Placement

• PDE-regularized placement: The Poisson energy, defined as the squared $H^{-1}$ norm of the placement density residual, acts as an analytically-justified, global balancing surrogate for overlap in geometric or chip layout optimization. It yields well-conditioned gradient flows, guarantees local linear convergence, and provides a principled link between energy minimization and geometric constraint satisfaction (Zhu et al., 9 Oct 2025).

4. Energy Conservation, Dissipation, and Structure Preservation

A core feature of energy-based models is the explicit relationship between the system dynamics and energy dissipation or conservation:

• Passivity: For any system with energy function $H$ and suitable structure/dissipation matrices, the generalized dissipation inequality

$$\frac{d}{dt} H \leq \langle \text{output}, \text{input} \rangle,$$

holds, ensuring energy cannot be created without external work (Altmann et al., 18 Jun 2024, Altmann et al., 17 Apr 2025).

• Thermodynamic laws: In physically consistent formulations (e.g., EPHS or free energy variational structures), the first and second law (energy conservation, nonnegative entropy production) are automatically inherited from the mathematical structure (Lohmayer et al., 13 Sep 2024, Gay-Balmaz et al., 2017). Models are closed under interconnection: submodels assembled using prescribed power/effort junctions result in aggregate models that maintain the same conservation and dissipation properties.
• Discrete stability and invariants: For numerical methods, exact or controlled preservation of the energy dissipation properties is possible via tailored time-stepping (midpoint rule, generalized-$\alpha$ method) and structure-preserving Petrov–Galerkin reductions (Liu et al., 2018, Altmann et al., 18 Jun 2024).

5. Design of Boundary Conditions, Parameterization, and Stability

Energy-based formulations enable robust and expressive boundary and interface treatments:

• Boundary conditions: Thermal and velocity boundary conditions in schemes like energy-conserving lattice Boltzmann, or energy-based hysteresis models, are formulated to match energy/momentum/entropy fluxes, resulting in physically meaningful wall responses and preserving conservation without spurious artifacts (Sbragaglia et al., 2010, Egger et al., 19 Jul 2025).
• Parameter selection and blending: In multiscale and coupling methods, energy-based error estimates lead to analytic choices of blending width, mesh grading, or oversampling radius that guarantee interface accuracy and uniform convergence rates, even across sharp heterogeneities (Luskin et al., 2011, Wang et al., 2020).
• Stability: Energy-based models yield precise timestep and mesh constraints for explicit (CFL) or implicit time discretizations, with grid refinement strategies guided by error propagation through the energy functional (Sbragaglia et al., 2010, Liu et al., 2018).

6. Impact, Limitations, and Interconnections

Energy-based formulations provide a systematic language connecting discrete, continuum, statistical, and data-driven models through a set of unifying principles:

• Modularity and extensibility: The energy function provides a modular abstraction for model assembly, enabling plug-and-play interconnection (EPHS, port-Hamiltonian, graph-based energy-asset models) and extensibility to new physics or technologies (Tejada-Arango et al., 2023, Lohmayer et al., 13 Sep 2024, Altmann et al., 18 Jun 2024).
• Physical interpretability: Losses and variational objectives correspond to true physical quantities or constraints, yielding results that are robust to architecture, choice of discretization, or problem setting (Kaba et al., 3 Nov 2025).
• Limitations: Not every physical or optimization problem admits an energy-based (global) structure; nonvariational dissipation or irreversibility (quantum jumps, certain stochastic processes) may resist a purely energy-based reduction. Moreover, interpretation of macroscopic energy densities or local force distributions can be fundamentally indeterminate in coarse-grained or averaged models (Westhoff, 12 Apr 2025).
• Broad influence: Applications range from optimizing power and engineering systems, understanding multiscale materials, governing turbulence transfers, learning generative models, to electronic design automation.

7. Representative Table: Energy-Based Formulation Types and Their Focus

Framework/Domain	Energy Functional Centrality	Key Conservation/Dissipation Guarantee
Volumetric LBM, FEM/PDE, CEM-GMsFEM	Governing equation, discrete conservation	Exact energy/mass/momentum conservation
Port-Hamiltonian, EPHS, generalized energy-based	Intrinsic model/DAE structure	Passivity, first/second law, structural interconnect.
Machine Learning (EBGAN, Boltzmann loss)	Loss function, score function	Manifold concentration, physically meaningful loss
PDE-based optimization (placement/floorplanning)	Regularization, surrogate constraint	Global balancing, overlap minimization
Atomistic-continuum and multiphysics coupling	Blended or modular system energies	Consistent transition, ghost-force minimization

The diversity of energy-based formulations across numerical methods, continuum physics, machine learning, and optimization reflects the centrality of energy minimization and conservation as both a modeling paradigm and an algorithmic design principle.