Compositional Energy Landscapes

• Compositional energy landscapes are frameworks that decompose complex, high-dimensional energy surfaces into modular subspaces, enabling tractable analysis and visualization.
• They employ mathematical methods like energy summation, manifold learning, and persistent homology to uncover structural and dynamic insights across diverse domains.
• Applications in molecular biophysics, materials science, and machine learning illustrate how modular representations drive efficient optimization and control of complex systems.

Compositional energy landscapes are mathematical and conceptual frameworks that decompose complex high-dimensional energy or free-energy surfaces into component subspaces or modules. Such landscapes arise in diverse domains, including molecular biophysics, statistical machine learning, combinatorial optimization, solid-state materials, and reasoning systems. The compositional perspective enables tractable analysis, visualization, and control of systems governed by myriad locally- and globally-interacting degrees of freedom. The following sections synthesize methodological advances and core findings from recent research, with focus on the rigorous modeling, representation, and practical exploitation of compositional energy landscapes.

1. Definition and Conceptual Foundations

A compositional energy landscape is a representation of a multidimensional energy (or free energy) function that is constructed, analyzed, or exploited by decomposing the complex system into constituent modules—such as molecular segments, subproblems, local descriptors, or symmetry-reduced configurations. The total energy or free energy $E(x)$ is then either written as a sum or composition of functions over these modules (e.g., $E(x) = \sum_k E_k(x_k)$), or studied via an explicit space of structured configurations (such as colored graphs, codebooks, or subproblem tuples).

This foundational modularity is central in domains such as:

• Molecular and materials science, where local and global energy contributions arise from chemical bonds, angles, torsions, solvation, and symmetry constraints (Russell et al., 2015, Jagatramka et al., 2022, Andreeva et al., 6 Nov 2024).
• Machine learning, where complex loss surfaces (energy functions on model spaces) are structured by the composition of algorithmic subcomponents or learned latent factors (Pavlovskaia et al., 2014, Du et al., 2021, Oarga et al., 23 Oct 2025).
• Disordered and glassy systems, which are naturally described as complex landscapes with competing basins/local minima shaped by interaction networks (Pusuluri et al., 2016, Po et al., 2021, Kahlke et al., 2023).

Compositionality extends to the integration of discrete (combinatorial) and continuous (geometric or physical) variables, and it supports both analysis (e.g., via landscape topologies or persistent homology (Mirth et al., 2020)) and the generation of algorithmically tractable representations for prediction and optimization.

2. Mathematical and Computational Frameworks

2.1. Energy Decomposition and Hierarchical Landscapes

In practice, compositional landscapes arise through explicit energy decompositions. For example:

• In supramolecular self-assembly, the total distortion energy of a polyhedral cage is written as a sum of geometric penalties for bond stretching, angle deviations, and vertex planarity, each governed by the local arrangement of ligands and metal ions. The configuration space is indexed by 2-colorings of the polyhedral edges, modulo the action of a symmetry group (Russell et al., 2015).
• In solid solutions, the total per-atom potential energy is reparameterized in terms of interactions among local coordination shells. Analytical expressions are derived for the average and statistical fluctuations of energy, expressed as sums over concentrations and pair interactions (Jagatramka et al., 2022).
• In machine learning, learned energy functions $E_\theta(x,y)$ are trained over the solution spaces of base subproblems. At test time, global landscapes are constructed as $E(\mathbf{y}) = \sum_k E^k_\theta(x_k, y_k)$, enabling compositional reasoning for harder, unseen instances (Oarga et al., 23 Oct 2025).
• In neural network models of memory and biological processes, the control and tuning of compositional landscapes is effected by adjusting the pairwise correlation structure among stored patterns, directly modulating basin size, density, and attractor hierarchy (Pusuluri et al., 2016).

2.2. Sampling and Visualization Strategies

Compositional structure is leveraged by specialized sampling, reduction, and visualization methods:

• For high-dimensional molecular or model landscapes, manifold learning techniques such as SHEAP (Stochastic Hyperspace Embedding and Projection) project structural minima or clusters onto low-dimensional spaces, preserving both compositional and energetic relationships (Shires et al., 2021).
• Persistent homology encodes the topological signature of sublevelsets across energy thresholds, allowing for rigorous quantification and comparison of landscape features such as basins, loops, and voids (Mirth et al., 2020).
• Energy Landscape Maps (ELMs) and disconnectivity graphs provide tree- or network-structured visualizations, in which leaves represent local minima, branches encode barrier heights, and associated measures like probability mass or volume annotate the relative significance of each compositional region (Pavlovskaia et al., 2014, Dobrynin et al., 2 Mar 2024).

Such methods facilitate both diagnostics (algorithmic performance, phase transitions, complexity) and actionable insights (e.g., which subproblems or modules dominate the global behavior and how their compositions affect solution quality or kinetics).

3. Applications in Physical, Biological, and Data Sciences

3.1. Protein Folding and Enzymatic Dynamics

Time-resolved, multi-temperature crystallography enables direct mapping of activation barriers and enthalpy/entropy decompositions for enzymatic reaction cycles, linking transient molecular structures to compositional energy profiles (Schmidt et al., 2013). This mapping informs mechanistic understanding, enzyme engineering, and rational drug design.

Markov state models, clustering, and persistent homology approaches yield compositional decompositions of high-dimensional conformational landscapes, revealing metastable core states and dynamic transition pathways (e.g., in calmodulin and other biomolecules) (Arbon et al., 2018, Westerlund et al., 2019).

3.2. Materials Science and Solid Solutions

Potential energy landscapes underpin material properties such as phase stability, fault tolerance, conductivity, and self-healing. Analytical composition-based models, calibrated and validated via atomistic simulations, now yield predictions of both average and local fluctuations in cohesive and defect energies as functions of local chemical environments (Jagatramka et al., 2022, Andreeva et al., 6 Nov 2024).

Visualization and mapping techniques can be applied to compositional systems ranging from binary alloys to quaternary chemical spaces, supporting the discovery and optimization of functional materials for energy storage, catalysis, and greenhouse gas capture (Shires et al., 2021, Andreeva et al., 6 Nov 2024).

3.3. Machine Learning and Combinatorial Optimization

Compositional energy landscape frameworks underpin advances in unsupervised concept discovery (e.g., learnable composable energy functions for visual scene attributes (Du et al., 2021)), global structure search in atomistic optimization (via construction of auxiliary smooth "complementary energy" landscapes for candidate generation (Slavensky et al., 28 Feb 2024)), and scalable reasoning (Oarga et al., 23 Oct 2025). Summing energies over learned subproblem modules enables generalization to more complex instances and seamless incorporation of additional constraints at inference.

Energy-based approaches are also central in theoretical and practical studies of computational hardness. Quantitative analysis of the disconnectivity and clustering in spin glasses, satisfiability, and polynomial-time matching problems highlights the conditions under which compositional landscapes become glassy or exhibit features analogous to replica symmetry breaking (Po et al., 2021, Kahlke et al., 2023).

4. Methodologies for Landscape Analysis and Inference

Methodology	Application Domain	Principal Function
Five-dimensional crystallography	Enzymatic kinetics, biophysics	Structural/thermodynamic barrier mapping
Generalized Wang–Landau algorithm	Non-convex statistical learning	Uniform landscape sampling, ELM construction
Gaussian mixture free energy estimates	Molecular simulation, biophysics	Automatic core state identification
Manifold learning embeddings	Cluster structure, materials	Dimensionality reduction, visualization
Persistent homology	Molecular dynamics, topology	Robust topological signature extraction
Combinatorial symmetry reduction	Polyhedral self-assembly, networks	Computational tractability
Parallel energy minimization (PEM)	Compositional reasoning, optimization	Particle-based nonconvex minimization
Graph neural Fokker-Planck (PESLA)	Stochastic dynamics, complex systems	Self-supervised energy estimation/inference

These methodologies are often deployed in combination to overcome sampling bottlenecks arising from high configurational dimensions and complex basin connectivity.

5. Control and Design of Compositional Energy Landscapes

Active control and rational design are achievable either via the explicit adjustment of compositional or modular properties (e.g., by tuning correlation matrices, local energy parameters, or subproblem energy functions (Pusuluri et al., 2016, Oarga et al., 23 Oct 2025)) or via symmetry-guided assembly protocols (Russell et al., 2015). In machine learning, compositional energy models support flexible system extension and fine-grained control over latent factors or constraints by modular addition of energy terms at inference, without retraining the entire global model (Du et al., 2021, Oarga et al., 23 Oct 2025).

In material systems, landscape engineering is realized by compositional optimization (e.g., solute selection in alloys to tune local fault stability (Jagatramka et al., 2022)), design of self-healing dielectrics via analysis of low-energy recovery paths (Andreeva et al., 6 Nov 2024), or synthetic modification of ligand geometries to direct supramolecular assembly (Russell et al., 2015).

6. Future Directions and Open Challenges

Recent research points to several promising developments:

• The integration of physics-informed, self-supervised machine learning (e.g., PESLA) for data-driven inference of compositional energy landscapes from time-series or trajectory data, even when ground-truth energies are unavailable (Li et al., 24 Feb 2025).
• Development of robust quantitative metrics—such as persistent homology distances—for comparing topological and compositional features of high-dimensional landscapes, enabling systematic benchmarking and optimization (Mirth et al., 2020).
• Advancements in scalable computational frameworks to construct, sample, and exploit compositional landscapes for ever-larger and more complex material, molecular, and reasoning problems (Andreeva et al., 6 Nov 2024, Oarga et al., 23 Oct 2025).

Challenges persist in ensuring the faithfulness of reduced or compositional models, managing the combinatorial explosion in configuration spaces for systems with high symmetry or compositional complexity, and in correlating compositional energy landscape features with macroscopic observables and performance metrics.

7. Representative Formulas and Notations

Some canonical landscape formulations include:

• Eyring transition state equation (biophysics):

$$k = \frac{k_B T}{h} \exp\left(\frac{\Delta S^\ddagger}{R}\right) \exp\left(-\frac{\Delta H^\ddagger}{RT}\right)$$

• Gaussian mixture density for free energy estimation:

$$p(x) \approx \sum_k w_k \mathcal{N}(x | \mu_k, \Sigma_k), \quad F(x) = -kT \ln p(x) + \text{const}$$

• Modular or compositional minimization (reasoning/optimization):

$$\hat{y} = \arg\min_y \sum_{k=1}^N E_\theta^k(x_k, y_k)$$

• Pairwise correlation structure modulating stability:

$$A^{\mu\nu} \equiv \frac{1}{N}\sum_i \xi_i^{(\mu)} \xi_i^{(\nu)}, \quad J_{ij}^{\text{KS}} = \frac{1}{N} \sum_{\mu,\nu} \xi_i^{(\mu)} (A^{-1})^{\mu\nu} \xi_j^{(\nu)}$$

• Fokker-Planck-based evolution on discrete landscape (PESLA):

$$\frac{\partial}{\partial t} \mathbf{H}_{c_i} = \sum_{j \in N(i)} \mathbf{W}_{ij}\left[E_{ji} + \beta_{\xi} (\log \mathbf{H}_{c_j} - \log \mathbf{H}_{c_i})\right] \circ \left[\sigma(kE_{ji})\,\mathbf{H}_{c_j} + (1-\sigma(kE_{ji}))\mathbf{H}_{c_i}\right]$$

These expressions, along with others detailed in the relevant literature, formalize the compositional, multi-scale, and modular architecture characteristic of energy landscapes in contemporary science and engineering.