Energy-Structured Causal Models (E-SCM)

• Energy-Structured Causal Models are causal frameworks that represent mechanisms as energy functions or vector fields, shifting focus from explicit input-output mappings to equilibrium states.
• They enable equilibrium selection and interventions by altering local energy constraints, thereby bridging traditional SCMs with energy-based and Bayesian modeling approaches.
• The framework facilitates practical applications such as energy demand prediction by aligning causal analysis with physically and behaviorally interpretable energy subsystems.

Searching arXiv for the specified papers and closely related work to ground the article. Energy-Structured Causal Models (E‑SCMs) are causal models in which mechanisms are represented as constraints, expressed either as energy functions or vector fields, rather than as explicit input–output assignments. In this formulation, system states are equilibria—typically minima of a total energy or fixed points of dynamics—and interventions are implemented as local surgeries on the corresponding constraints, followed by re-equilibration (Thomas, 24 Oct 2025). Within this broad formulation, the demand-forecasting model in "Causal Inference in Energy Demand Prediction" functions as a domain-specific instance of an E‑SCM: variables, edges, and structural equations are organized around interpretable energy subsystems such as HVAC needs, routine activity needs, and lighting needs, and the resulting structure is used both for causal analysis and Bayesian prediction (Ma et al., 12 Dec 2025).

1. Formal definition and core semantics

A static E‑SCM is defined over endogenous latent variables $\mathbf Z=(Z_1,\dots,Z_n)$ and exogenous variables $\mathbf U=(U_1,\dots,U_n)$, together with a DAG $\mathcal G$ specifying parent sets $\mathrm{PA}(i)$, local mechanism energies $\{E_i\}_{i=1}^n$, exogenous energies $\{E_{U_i}\}_{i=1}^n$, and an optional global term $E_{\mathrm{global}}$ (Thomas, 24 Oct 2025). The local mechanisms satisfy

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$

while exogenous energies satisfy

$E_{U_i}:\ \mathcal U_i\to\mathbb R.$

The total energy is

$$E(\mathbf z,\mathbf u) = \sum_{i=1}^n E_i\bigl(z_i\mid z_{\mathrm{PA}(i)},u_i\bigr) + \sum_{i=1}^n E_{U_i}(u_i) + E_{\mathrm{global}}(\mathbf z,\mathbf u).$$

Admissible states are equilibria satisfying

$\mathbf U=(U_1,\dots,U_n)$0

typically under convexity or uniqueness conditions ensuring a unique equilibrium for each exogenous context (Thomas, 24 Oct 2025).

A dynamic E‑SCM replaces scalar energies with local vector fields,

$\mathbf U=(U_1,\dots,U_n)$1

so that mechanisms are differential constraints and interventions operate by rewriting selected local dynamics (Thomas, 24 Oct 2025). Static and dynamic forms coincide when the dynamics are gradient flows, $\mathbf U=(U_1,\dots,U_n)$2, but the dynamic formalism also admits non-gradient systems.

The central semantic shift is from explicit structural assignments to equilibrium selection under constraints. In a classical SCM, a mechanism computes $\mathbf U=(U_1,\dots,U_n)$3 from $\mathbf U=(U_1,\dots,U_n)$4 and $\mathbf U=(U_1,\dots,U_n)$5. In an E‑SCM, the mechanism defines which joint configurations are preferred or admissible, and the realized state is determined by global equilibrium across all such constraints (Thomas, 24 Oct 2025). This makes the unit of intervention an energy term or vector-field component rather than a structural equation in explicit solved form.

2. Relation to classical SCMs

The formal relationship between E‑SCMs and standard SCMs is established through a reduction theorem. For an E‑SCM with energy

$\mathbf U=(U_1,\dots,U_n)$6

under locality, blockwise strict convexity, global strict convexity and coercivity, modular interventions, exogenous stability, and measurability, one defines local best-response maps

$\mathbf U=(U_1,\dots,U_n)$7

The unique global minimizer $\mathbf U=(U_1,\dots,U_n)$8 then satisfies

$\mathbf U=(U_1,\dots,U_n)$9

and conversely any solution of

$\mathcal G$0

is the unique minimizer of the energy (Thomas, 24 Oct 2025).

The induced SCM has structural equations

$\mathcal G$1

and the same exogenous law $\mathcal G$2. Under the theorem’s assumptions, the E‑SCM and induced SCM are observationally equivalent, interventionally equivalent, and counterfactually equivalent (Thomas, 24 Oct 2025). This means that E‑SCMs do not discard SCM semantics; rather, they reparameterize those semantics in a constraint-based form.

This equivalence is important for causal inference because it preserves the standard abduction–intervention–prediction interpretation. In E‑SCM form, abduction corresponds to constrained energy minimization to infer exogenous variables and latent states; intervention corresponds to editing local energies; prediction corresponds to re-solving for equilibrium in the edited system (Thomas, 24 Oct 2025). A plausible implication is that E‑SCMs are best understood not as a rejection of SCMs, but as an alternative mechanism representation particularly suited to latent-variable and equilibrium-based settings.

3. Mechanisms, interventions, and counterfactuals

In E‑SCMs, mechanisms compose additively:

$\mathcal G$3

Because mechanisms are separate energy terms, adding, removing, or editing a mechanism is literally addition, deletion, or replacement of terms in the total energy (Thomas, 24 Oct 2025). This additive structure is the energy analogue of modular decomposition in SCMs.

Hard interventions $\mathcal G$4 are implemented by deleting the original local mechanism for $\mathcal G$5 and imposing an infinite barrier outside the intervened value:

$\mathcal G$6

Soft interventions replace the local term by a convex combination of original and intervened energies,

$\mathcal G$7

with $\mathcal G$8 (Thomas, 24 Oct 2025). The framework also includes disjunctive interventions $\mathcal G$9 either as policy-free families of singleton interventions or by introducing a control energy over admissible values.

Counterfactuals follow the standard three-step structure. First, abduction solves

$\mathrm{PA}(i)$0

Second, intervention edits the relevant local energies while holding the abducted exogenous variables fixed. Third, prediction re-minimizes the edited energy to obtain post-intervention outcomes (Thomas, 24 Oct 2025). The formal identity with SCM counterfactual semantics holds under the reduction theorem’s assumptions.

4. Structural principles: LAP, ICM, and gauge ambiguity

A distinctive feature of the E‑SCM program is that structural principles are expressed as differential conditions on energies or vector fields. The Locality-Autonomy Principle (LAP) requires that a node’s mechanism not depend on non-descendants, either through their states or through their parameters. For a static E‑SCM, if $\mathrm{PA}(i)$1 denotes the effective energy for node $\mathrm{PA}(i)$2 relative to a module $\mathrm{PA}(i)$3, then for non-descendant $\mathrm{PA}(i)$4 the required conditions are

$\mathrm{PA}(i)$5

Training can penalize violations through

$\mathrm{PA}(i)$6

(Thomas, 24 Oct 2025).

Independent Causal Mechanisms (ICM) is formulated in parameter space. Defining the residual

$\mathrm{PA}(i)$7

the conditions are

$\mathrm{PA}(i)$8

Corresponding penalties are introduced to enforce approximate ICM during learning (Thomas, 24 Oct 2025). CAP, the Compositional Autonomy Principle, is treated as an outlook principle concerning transport of mechanism templates and preservation of intervention responses across domains.

The E‑SCM literature also uses a gauge-theoretic analysis of encoder–energy pairs to explain why empirical risk minimization can yield causally opaque internal representations. If $\mathrm{PA}(i)$9 is an encoder and $\{E_i\}_{i=1}^n$0 an energy on the latent space, then gauge transformations of the form

$\{E_i\}_{i=1}^n$1

can preserve observational heads while altering intervention-relevant internal structure (Thomas, 24 Oct 2025). The paper distinguishes the larger set of transformations preserving observational heads from the smaller set preserving causal heads, and interprets the difference as the locus of fractured or entangled representations. Probes such as absolute per-module energies, gradients, energy differences, and Hessians progressively shrink this observational gauge freedom.

A plausible implication is that E‑SCMs attempt to make causal modularity directly inspectable and regularizable at the representation level, not solely at the level of externally observed statistical relationships.

5. Energy demand prediction as a worked example

"Causal Inference in Energy Demand Prediction" presents a structural causal model for load forecasting that can be read as a concrete E‑SCM for an energy system (Ma et al., 12 Dec 2025). The observed exogenous root variables are calendar variables hour of day $\{E_i\}_{i=1}^n$2 and month of year $\{E_i\}_{i=1}^n$3. Observed weather variables are temperature $\{E_i\}_{i=1}^n$4, relative humidity $\{E_i\}_{i=1}^n$5, wind speed $\{E_i\}_{i=1}^n$6, and solar radiation $\{E_i\}_{i=1}^n$7. Latent or conceptual variables are HVAC needs, routine activity needs, and lighting needs. The target is energy demand $\{E_i\}_{i=1}^n$8.

The proposed DAG organizes these variables into physically and behaviorally meaningful subsystems. Calendar variables affect weather; weather affects HVAC needs; calendar variables affect routine activities and hence routine activity needs; calendar variables and solar radiation affect lighting needs; and the three need-based subsystems combine to determine total demand (Ma et al., 12 Dec 2025). In conceptual structural-equation form,

$\{E_i\}_{i=1}^n$9

The joint distribution factorizes as

$\{E_{U_i}\}_{i=1}^n$0

Temperature is modeled conditionally on hour and month through a harmonic series:

$\{E_{U_i}\}_{i=1}^n$1

with Gaussian noise (Ma et al., 12 Dec 2025). Relative humidity, solar radiation, and wind speed are assigned Gaussian priors in the Bayesian implementation; the paper explicitly treats this as a modeling simplification rather than a causal claim of independence from calendar variables.

The temperature response of demand is represented through a V-shape transform

$\{E_{U_i}\}_{i=1}^n$2

with $\{E_{U_i}\}_{i=1}^n$3 (Ma et al., 12 Dec 2025). Base HVAC demand is then

$\{E_{U_i}\}_{i=1}^n$4

Humidity contributes conditionally in the hot regime:

$\{E_{U_i}\}_{i=1}^n$5

where $\{E_{U_i}\}_{i=1}^n$6 is chosen as $\{E_{U_i}\}_{i=1}^n$7 by grid search maximizing correlation with demand. Wind contributes asymmetrically across thermal regimes:

$\{E_{U_i}\}_{i=1}^n$8

Daily activity demand is modeled as an hourly harmonic term,

$\{E_{U_i}\}_{i=1}^n$9

and yearly activity demand as a monthly harmonic term,

$E_{\mathrm{global}}$0

Lighting demand is

$E_{\mathrm{global}}$1

with the active-hour indicator equal to 1 between 5 a.m. and 12 a.m. and 0 otherwise (Ma et al., 12 Dec 2025).

Total demand is decomposed additively as

$E_{\mathrm{global}}$2

This additive subsystem structure is the sense in which the model is "energy-structured" in domain terms: functions are aligned to interpretable physical and behavioral mechanisms rather than to a purely correlational predictor (Ma et al., 12 Dec 2025).

6. Causal identification, Bayesian implementation, and empirical behavior

The demand paper emphasizes confounding explicitly. For humidity, the DAG contains the paths

$E_{\mathrm{global}}$3

Thus $E_{\mathrm{global}}$4 is a confounder of humidity and energy demand in Pearl’s sense (Ma et al., 12 Dec 2025). A naive regression yields

$E_{\mathrm{global}}$5

which is inconsistent with the domain expectation that higher humidity increases cooling demand. Conditional on hour and temperature, the sign becomes positive in hot regimes. The paper interprets this as a positive direct causal path from humidity to demand together with a negative backdoor path through hour.

For temperature, the target causal quantity is $E_{\mathrm{global}}$6 with $E_{\mathrm{global}}$7, $E_{\mathrm{global}}$8, and $E_{\mathrm{global}}$9. The paper states that $$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$0 satisfies the backdoor criterion for $$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$1, so

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$2

If

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$3

with $$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$4, then

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$5

Accordingly, the adjusted regression coefficient $$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$6 coincides with the causal coefficient in

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$7

The paper contrasts this causal adjustment with a non-causal approach that regresses on temperature and then separately on the daily cycle; the latter is reported to produce biased temperature coefficients (Ma et al., 12 Dec 2025).

The Bayesian model operationalizes the causal structure by placing priors over mechanism parameters and using stochastic variational inference in Pyro with an AutoNormal guide (Ma et al., 12 Dec 2025). For each hour-indexed datum,

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$8

The paper provides examples of Gaussian priors such as

$$E_i:\ \mathcal Z_i\times \mathcal Z_{\mathrm{PA}(i)} \times \mathcal U_i \to \mathbb R,$$9

with $E_{U_i}:\ \mathcal U_i\to\mathbb R.$0.

Performance is evaluated using Mean Absolute Percentage Error,

$E_{U_i}:\ \mathcal U_i\to\mathbb R.$1

The reported results are train MAPE $E_{U_i}:\ \mathcal U_i\to\mathbb R.$2, test MAPE $E_{U_i}:\ \mathcal U_i\to\mathbb R.$3, and 5-fold cross-validation average MAPE $E_{U_i}:\ \mathcal U_i\to\mathbb R.$4 across two years of data (Ma et al., 12 Dec 2025). The paper also reports that ignoring confounding yields 47.8% deviation in temperature coefficients and 12.5% worse out-of-sample MAPE than the causal approach.

The model’s seasonal analysis attributes lower winter variance in demand to decoupling between temperature-driven HVAC fluctuations and activity-driven fluctuations. In summer, midday heat and midday activity are aligned, producing positive covariance between subsystem peaks; in winter, heating demand peaks in morning and evening while activity remains concentrated around midday, so the covariance is lower (Ma et al., 12 Dec 2025). The paper treats this heteroscedasticity as emergent from the causal structure rather than as a separately fitted variance model.

7. Position in causal machine learning and generalization across energy systems

E‑SCMs occupy an intermediate position between SCMs, energy-based models, dynamical systems, and domain-structured Bayesian models. Relative to SCMs, they are more general in representation but, under mild assumptions, equivalent in causal semantics (Thomas, 24 Oct 2025). Relative to graphical models, they factor a deterministic energy rather than a probability distribution. Relative to ordinary energy-based models, they add a DAG, explicit intervention semantics, and diagnostic principles such as LAP and ICM. Relative to unconstrained deep learning, they aim to provide mechanism-level editability and causal interpretability rather than observational adequacy alone (Thomas, 24 Oct 2025).

The foundational paper is mainly conceptual and does not present full empirical case studies, but it sketches an architectural ecosystem comprising adaptors, mechanisms, actuators, and probes. Adaptors map raw neural representations into causal latents and provide initializers for equilibrium solves. Mechanisms are parametric energies or vector fields with DAG-based masking and LAP/ICM penalties. Actuators perform intervention by editing specific local energies. Probes expose internal quantities such as equilibria, energies, gradients, and Hessians for diagnostics and evaluation (Thomas, 24 Oct 2025). Training combines a primary observational objective with structural regularizers.

The demand-prediction paper suggests how this outlook can be specialized to energy applications. Its construction pipeline proceeds from domain knowledge to a causal DAG, from DAG to structural equations, and from structural equations to a Bayesian predictive model (Ma et al., 12 Dec 2025). The same document proposes broader construction steps for other energy domains: define physical and behavioral subsystems; list exogenous drivers, latent process states, and outputs; draw the causal DAG; specify mechanism-reflective structural equations with piecewise or nonlinear effects; encode the model as a Bayesian program; and validate causal structure through conditional independencies, backdoor reasoning, and causal-versus-non-causal comparisons.

Examples mentioned include renewable generation, industrial loads, and EV charging (Ma et al., 12 Dec 2025). In these applications, the E‑SCM perspective would retain the same organizing principle: variables and functional forms are arranged around physically and institutionally meaningful mechanisms, interventions are evaluated as edits to those mechanisms, and robustness under non-stationarity is sought through structural rather than purely correlational inductive bias. This suggests that the term "Energy-Structured Causal Model" has two closely related uses in the current literature: a general causal formalism based on energies and equilibria (Thomas, 24 Oct 2025), and a domain-specific modeling stance in which causal structure is organized around interpretable subsystems of an energy application (Ma et al., 12 Dec 2025).