Potential Energy-Based Modeling

• Potential energy-based modeling is a framework that uses scalar potentials to encode interactions, constraints, and stability across diverse fields.
• It employs methods like elastic potential declines, potential flows, and energy matching to refine representations and generate accurate predictive models.
• Applications include improved transfer learning correlations, enhanced generative performance, precise molecular simulations, and robust control strategies.

Potential energy-based modeling denotes a family of formulations in which a scalar potential energy, a potential energy surface, or a desired closed-loop potential is used to encode interactions, likelihood structure, constraints, or stability objectives. In the cited literature, this role appears in transfer learning as a surrogate for fine-tuning dynamics, in generative modeling as the energy of a Boltzmann density or the potential underlying a conservative flow, in atomistic and molecular science as an inter-atomic or molecular potential energy surface, and in control and robotics as the object to be shaped or directly formulated in actuation space (Li et al., 2023, Loo et al., 2024, Zhang et al., 2018, Harandi et al., 2021).

1. Core meanings of potential energy-based modeling

The term is not used in a single uniform sense. In some work, potential energy is a physically inspired surrogate over latent representations; in other work, it is the scalar energy of an energy-based probabilistic model; in molecular modeling it is the inter-atomic or molecular potential energy surface; and in control it is the desired or actuation-induced energy whose shaping yields stability or tractable dynamics.

Domain	Role of potential energy	Representative preprints
Transfer learning	Elastic potential over overlapping class clusters in latent space	(Li et al., 2023)
Generative modeling	Scalar energy defining a Boltzmann density or a potential flow	(Loo et al., 2024, Loo et al., 22 Apr 2025, Balcerak et al., 14 Apr 2025)
Molecular and materials modeling	Inter-atomic or reduced-dimensional potential energy surface	(Han et al., 2017, Zhang et al., 2018, Ding, 19 Jul 2025)
Control and robotics	Desired potential for energy shaping or actuation-space potential	(Harandi et al., 2021, Yang et al., 15 Dec 2025)

A recurring distinction is between physically literal and formally abstract uses of energy. The Deep Potential and DeepPot-SE lines model the potential energy surface of atomic and molecular systems directly, preserving natural symmetries and extensivity (Han et al., 2017, Zhang et al., 2018). By contrast, potential-flow EBMs and energy matching learn a scalar field whose equilibrium or gradient field defines a probability model, while PED treats class clusters as repelling balls and uses elastic potential decline as a proxy for post-fine-tuning representation evolution (Loo et al., 2024, Balcerak et al., 14 Apr 2025, Li et al., 2023).

2. Representation dynamics and transferability estimation

In transfer learning, the PED method reframes model selection through the lens of potential energy decline. The motivating claim is that existing transferability measures such as LogME, GBC, and SFDA rely on static feature-label correlations and overlook the dynamic changes of representations during fine-tuning, which is especially unreliable for self-supervised models (Li et al., 2023). PED models each class embedding distribution as a multivariate Gaussian $\mathcal{N}(c_i,\sigma_i^2)$, then simplifies it to a ball with center $c_i$ and radius $r_i=\lambda\|\sigma_i\|_2$. For overlapping class balls, the pairwise potential energy is modeled after Hooke’s law:

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$

and the total system energy is

$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$

The associated force model is likewise elastic. For overlapping clusters,

$$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$

with acceleration $\boldsymbol{a}_i=\boldsymbol{F}_i$ when $m_i=1$, and a simulated representation update

$$\overline{Z_i}=Z_i+\frac{1}{2}\boldsymbol{a}_i\Delta t^2.$$

This update is iterated until the system’s potential energy converges, either when force falls below a threshold $\epsilon$ or when a maximum number of steps is reached. The intent is not to fine-tune the network itself, but to generate a refined feature configuration that reflects likely post-fine-tuning dynamics.

PED is explicitly plug-and-play: PED + LogME, PED + GBC, and PED + SFDA are all evaluated. Across 10 downstream tasks and 12 self-supervised models, the PED-augmented variants consistently outperform their vanilla counterparts in weighted Kendall’s $c_i$0 correlation with actual transfer rankings. The reported examples include LogME improving from 0.223 to 0.509 on Aircraft and from -0.008 to 0.667 on Cifar100 after PED. Qualitatively, t-SNE visualizations show self-supervised features that are chaotically mixed before PED and more distinct after the simulation, resembling the effect of actual fine-tuning. The method is also described as computationally lightweight because it requires only class-wise statistics and a small number of iterations (Li et al., 2023).

3. Energy-based probability, potential flows, and transport-aligned generation

A large branch of potential energy-based modeling is probabilistic. In the standard EBM form, the density is specified by a scalar energy, for example

$c_i$1

or, in the sign convention used by some potential-flow papers,

$c_i$2

The main difficulty identified across these works is that conventional training often relies on implicit MCMC sampling during contrastive divergence, which is unstable and expensive in high dimensions (Loo et al., 2024, Loo et al., 22 Apr 2025).

VAPO replaces MCMC-based sampling with a learned potential flow. It learns a potential energy function $c_i$3 whose gradient transports prior samples so that their density evolution matches an approximate data likelihood homotopy. The dynamics are written as

$c_i$4

and the associated density evolution obeys

$c_i$5

Training minimizes the time-integrated KL divergence between the flow-driven density and the target homotopy, using a variational loss derived from a weighted Poisson equation. After training, sampling is performed by initializing from a Gaussian prior and solving the ODE on a fixed time interval. Reported unconditional image-generation results include FID 16.6 on CIFAR-10 and 14.5 on CelebA (64x64) (Loo et al., 2024).

VPFB extends this line by constructing a flow-driven density homotopy matched to the data distribution through a variational loss minimizing the KL divergence between the flow-driven and marginal homotopies. The flow is again conservative,

$c_i$6

and the matching condition leads to a density-weighted Poisson equation

$c_i$7

The paper casts this in a Deep Ritz variational form and states that, in the stationary regime, the learned potential energy coincides with the EBM energy through a Boltzmann distribution $c_i$8 (Loo et al., 22 Apr 2025).

Energy Matching proposes a single time-independent scalar potential $c_i$9 that interpolates between an optimal-transport regime far from the data manifold and a Boltzmann equilibrium regime near the data manifold. Its organizing equation is the JKO step

$r_i=\lambda\|\sigma_i\|_2$0

When $r_i=\lambda\|\sigma_i\|_2$1, trajectories follow curl-free optimal transport paths; when entropy is turned on, the equilibrium density becomes

$r_i=\lambda\|\sigma_i\|_2$2

The reported CIFAR-10 FID is 3.97, with explicit emphasis that the same scalar potential remains composable with additional energy terms for inverse problems and interaction energies (Balcerak et al., 14 Apr 2025).

A related sampling problem is addressed in “Particle Dynamics for Learning EBMs,” which avoids crude MCMC by explicitly matching the evolution of samples with the evolution of the energy model. The main vector field result is

$r_i=\lambda\|\sigma_i\|_2$3

so particles are transported deterministically as the energy changes. The discrete update in method $r_i=\lambda\|\sigma_i\|_2$4 is

$r_i=\lambda\|\sigma_i\|_2$5

The paper reports that its deterministic particle updates capture all modes more faithfully than Persistent Contrastive Divergence on synthetic experiments (Neklyudov et al., 2021).

The statistical role of the energy parameterization itself is analyzed in the shallow-network EBM literature. There, an EBM is written as

$r_i=\lambda\|\sigma_i\|_2$6

with shallow neural networks used as energy approximators. The paper distinguishes an active regime, in which both features and output weights are learned, from a lazy or kernel regime, in which features are fixed. For data distributions with low-dimensional structure, the active regime yields KL convergence rates of order $r_i=\lambda\|\sigma_i\|_2$7, where $r_i=\lambda\|\sigma_i\|_2$8 is the intrinsic rather than ambient dimension, while the lazy regime does not exploit the same structure (Domingo-Enrich et al., 2021).

4. Structured prediction, conditional inference, and discrete domains

Potential energy-based modeling has also been extended to structured and partially observed data. In $r_i=\lambda\|\sigma_i\|_2$9-EBM for functional data, the target object is a distribution over functions rather than over finite-dimensional vectors. The construction starts from a Gaussian Process spectral decomposition,

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$0

and then introduces an energy-tilted latent prior

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$1

Because the function is represented in the spectral basis, the model can use irregularly sampled training data and produce predictions at any desired resolution. The paper reports strong performance on synthetic data, S&P 500, UK National Grid, Melbourne pedestrian counts, and FashionMNIST treated as functions (Lim et al., 2022).

In multi-agent robotics, the Energy-based Potential Game formulation connects differential games, optimal control, and EBMs. For discrete time, the energy-minimizing strategy is written as

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$2

with neural networks inferring the parameters $$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$3 from observations and a differentiable game-theoretic optimization layer solving the resulting problem. The paper states that this layer adds interpretability and improves predictive performance on two simulations and two real-world driving datasets, with lower scene-level metrics such as minSADE and minSFDE and a nearly factor of 4 reduction in minSADE on CARLA relative to a strong baseline (Diehl et al., 2023).

For noisy-label learning, PEMM uses a distance-based classifier together with potential energy regularization on class centers. The classifier uses

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$4

and regularizes the class centers through

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$5

The complete objective is

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$6

Experiments on CIFAR-10, CIFAR-100, and Clothing1M report state-of-the-art performance, including 87.9% test accuracy for PEMM versus 78.7% for CE at 40% noise and 72.8% accuracy on Clothing1M (Wang et al., 2024).

Potential energy is also used for multimodal scientific diagnostics and discrete graph generation. The LAPD plasma EBM defines

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$7

with a CNN- and attention-based architecture over 699 features comprising diagnostics, machine settings, and probe positions. It supports conditional reconstruction through an auxiliary energy

$$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$8

and is used for diagnostic reconstruction, probe-position inference by direct energy evaluation, conditional sampling over machine inputs, and unconditional multimodal generation (Travis et al., 9 May 2026). In graph generation, GEM learns a permutation-invariant potential $$U_{ij}=\frac{1}{2}k x_{ij}^2,\qquad x_{ij}=\max(r_i+r_j-d_{ij},0),$$9 and combines a JKO-motivated transport step with a locally balanced mixing regime, enabling conditional generation, property-constrained sampling, and geodesic interpolation while matching or exceeding strong discrete diffusion baselines on molecular graph benchmarks (Balcerak et al., 24 Mar 2026).

5. Physical, molecular, and biological potential surfaces

In atomistic and molecular science, potential energy-based modeling often means direct approximation of a potential energy surface. Deep Potential represents the total energy as a sum of local atomic contributions,

$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$0

with each atomic sub-network depending only on neighbors within a cutoff radius and with symmetry enforced by deterministic preprocessing in local reference frames. The method is reported to reproduce underlying empirical or quantum-mechanical models within chemical accuracy, with examples including 0.04 eV MAE on C$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$1O$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$2H$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$3, 0.20 meV/atom RMSE for Cu, 0.09 meV/atom for Zr, and 1.8 meV/mol for H$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$4O, together with practical $$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$5 scaling and favorable transfer to larger Cu systems (Han et al., 2017).

DeepPot-SE refines this line into an end-to-end, symmetry-preserving, extensive, and continuously differentiable PES model. For coordinates $$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$6 it writes

$$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$7

where $$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$8 is the local environment of atom $$U(Z)=\sum_i\sum_{j\neq i}\frac{1}{2}k\max(r_i+r_j-d_{ij},0)^2.$$9. The local coordinates are weighted by a smooth cutoff function

$$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$0

which ensures continuous differentiability of energies, forces, and virials. The paper states that DeepPot-SE describes finite and extended systems including organic molecules, metals, semiconductors, and insulators with high fidelity (Zhang et al., 2018).

A more explicitly quantum-mechanical use of potential energy appears in the hydrogen-like atom model for interatomic interactions. There the Fourier component of the interaction energy is

$$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$1

which is quartic in the normalized atomic form factors. Taking the Pauli principle into account leads to two local potential wells separated by a potential barrier. For the beryllium dimer, the model gives satisfactory agreement with experiment in the depth of the potential well, while its position and width differ, and a new deeper minimum at shorter separation is predicted but described as not stable under normal conditions (Koshcheev et al., 2020).

Reduced-dimensional PES modeling is used in time-resolved photodissociation of diiodomethane. The model employs two degrees of freedom for photodissociation, $$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$2 and $$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$3, and three for the subsequent Coulomb explosion, with the ionic surface written as

$$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$4

The resulting photodissociation paths are consistent with previous ab initio molecular dynamics simulations and indicate the $$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$5 rotational period to be around 340 fs. Theoretical time-dependent kinetic energy release and angle correlations show good agreement with experiment in parts, confirming the static $$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$6 state and the $$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$7 dissociation channels (Ding, 19 Jul 2025).

In protein refinement, the tFold-TR framework uses a hybrid potential energy function

$$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$8

combining template-derived restraints with deep learning-predicted distances for missing regions and residue pairwise-specific deviations for variable restraint strength. For aligned regions,

$$\boldsymbol{F}_{ij}=k x_{ij}\cdot\boldsymbol{n},\qquad \boldsymbol{F}_i=\sum_{j\neq i}\boldsymbol{F}_{ij},$$9

with $\boldsymbol{a}_i=\boldsymbol{F}_i$0 set from the predicted deviation. The reported CASP13TBM result improves GDT-TS from 66.0 for the baseline TBM model to 74.8 for predicted-delta tFold-TR (Zheng et al., 2021).

6. Energy shaping, constrained mechanics, and actuation-space formulations

In control of underactuated mechanical systems, potential energy-based modeling appears as potential energy shaping. Within IDA-PBC, the relevant matching PDE is

$\boldsymbol{a}_i=\boldsymbol{F}_i$1

and the homogeneous part is

$\boldsymbol{a}_i=\boldsymbol{F}_i$2

The reformulation paper shows that, under a verifiable condition at the desired equilibrium,

$\boldsymbol{a}_i=\boldsymbol{F}_i$3

stabilization may be obtained by solving only the homogeneous part. When $\boldsymbol{a}_i=\boldsymbol{F}_i$4 is chosen in quadratic form, the condition can be reduced to an LMI, with an explicit 2-DOF form

$\boldsymbol{a}_i=\boldsymbol{F}_i$5

The paper applies this reformulation to an underactuated spatial cable-driven robot, Acrobot, Pendubot, Cart-Pole, and VTOL aircraft (Harandi et al., 2021).

A broader alternative to the port-Hamiltonian framework is proposed in “A novel energy-based modeling framework.” Its core system is

$\boldsymbol{a}_i=\boldsymbol{F}_i$6

with output

$\boldsymbol{a}_i=\boldsymbol{F}_i$7

Here $\boldsymbol{a}_i=\boldsymbol{F}_i$8 is not included in the energy and serves to encode constraints and algebraic relations directly. The framework preserves the dissipation inequality $\boldsymbol{a}_i=\boldsymbol{F}_i$9, is closed under structure-preserving interconnection, admits Petrov-Galerkin projection, and supports midpoint-rule and discrete-gradient time integrators that preserve dissipation. The ten examples include poroelasticity, index-1 and index-2 DAEs, circuit models, constrained mechanical systems, a vibrating string, a viscoelastic Stokes problem, and the Cahn–Hilliard equation (Altmann et al., 2024).

For cable-driven continuum robots, LASEM formulates actuation potential energy directly in actuation space. With cable length change

$m_i=1$0

the force-input actuation energy becomes

$m_i=1$1

A variational derivation reduces the governing dynamics to a single PDE,

$m_i=1$2

so that only the Euler moment balance is required while the Newton force balance is incorporated implicitly. The paper emphasizes that the formulation avoids explicit cable-backbone contact-force computation, supports both force-input and displacement-input modes, and, with Galerkin space-time modal discretization, yields an average 62.3% computational speedup over state-of-the-art real-time dynamic modeling approaches (Yang et al., 15 Dec 2025).

7. Distinctions, limitations, and open directions

A central misconception is that “potential energy” refers to a single formal object across all of these papers. The corpus shows otherwise. In PED it is an elastic analogy defined on class clusters in latent space; in VAPO, VPFB, Energy Matching, GEM, and the plasma EBM it is the scalar energy of a probabilistic model; in Deep Potential, the hydrogen-like atomic interaction model, and reduced-dimensional photodissociation it is a physical potential energy surface; and in IDA-PBC and LASEM it is either a desired closed-loop potential or an actuation-space energy (Li et al., 2023, Loo et al., 2024, Zhang et al., 2018, Harandi et al., 2021).

The literature also records nontrivial limitations. The hydrogen-like atomic interaction model matches the well depth of the beryllium dimer satisfactorily, but its well position and width differ from experiment (Koshcheev et al., 2020). The reduced-dimensional diiodomethane model does not include surface hopping or non-adiabatic transitions and does not explain lower KER bands that may correspond to three-body breakups or rare isomerizations (Ding, 19 Jul 2025). In tFold-TR, the native-delta strategy is presented as an upper-limit benchmark, indicating that final accuracy depends on the quality of deviation prediction (Zheng et al., 2021). In plasma analysis, the EBM is explicitly valuable because a single trained model can reconstruct diagnostics, solve inverse problems, and support anomaly-related exploration, but it still relies on Langevin dynamics and replay-buffer training (Travis et al., 9 May 2026).

This suggests that potential energy-based modeling is best understood not as a single formalism but as a modeling pattern: encode geometry, interaction, stability, or likelihood in a scalar energy and use gradients, variational principles, equilibrium distributions, or dissipation laws to infer states, trajectories, or samples. The cited work further suggests that current research is moving in two directions simultaneously: away from simulation-heavy sampling in generative EBMs through potential flows and transport-aligned objectives, and toward more explicit exploitation of energy structure in application domains where constraints, symmetry, and interpretability are central (Loo et al., 22 Apr 2025, Balcerak et al., 14 Apr 2025, Yang et al., 15 Dec 2025).