Phase Transition–Inspired Losses in Learning

• Phase transition–inspired losses are objective functions that embed statistical physics principles to produce abrupt changes in model complexity and fitting behavior.
• They utilize Landau-style polynomial potentials and free energy analogies to regulate model selection, yielding continuous or discontinuous transitions in learned representations.
• Applications range from enhancing implicit neural representations in surface reconstruction to informing regularization in non-equilibrium quantum and caloric systems.

Phase transition–inspired losses are a class of objective functions for learning and optimization that explicitly encode statistical-mechanical phase transition phenomena into the landscape of model selection and fitting. These losses are designed to induce—through their parameter and architecture dependence—sharp transitions in the learned model characteristics, mirroring the abrupt changes observed in physical systems at critical points. The principle arises both from precise mathematical analogies to statistical mechanics and from the formal structure of phase transitions found in fluid dynamics, condensed matter systems, and complex information-theoretic models. Constructing and analyzing such losses enables principled regularization, minimality, or robustness properties that are unattainable with classical empirical risk minimization and conventional regularizers.

1. Theoretical Foundations and Analogies

The statistical physics formalism grounds phase transition–inspired losses. In the prototypical construction, the loss is viewed as a "free energy" functional analogous to $F = U - T S$ in thermodynamics, balancing a "prediction error" (internal energy) and a "complexity penalty" (entropic or regularization term) controlled by a hyperparameter that plays the role of temperature. The competition between fit quality and complexity generates a sharp "phase transition" as the regularization parameter crosses a threshold: the model abruptly shifts from an "ordered" (structured, low-complexity) to a "disordered" (unstructured, high-complexity) regime (Ziyin et al., 2022).

This analogy is mathematically operationalized by functions such as

$\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$

where $\ell_0$ is the data fit term, $R(w)$ is a regularization or complexity proxy, and $a$ is a scaling parameter analogous to inverse temperature. In limiting regimes, minimization of such losses yields transitions replacing smooth parameter response with discontinuous or singular behavior in model properties.

2. Prototype Loss Functions and Phase Transition Manifestations

A canonical form of phase transition–inspired loss employs a Landau-style polynomial potential in the norm of parameters,

$$R(w) = \lambda_2 \|w\|_2^2 - \lambda_4 \|w\|_2^4 + \lambda_6 \|w\|_2^6$$

with $\lambda_2$ swept through zero. This structure induces:

• Second-order (continuous) phase transitions for shallow networks (e.g., one hidden layer), with critical exponents matching mean-field theory and continuous emergence of nontrivial solutions as the regularization parameter passes the threshold $\gamma_c$.
• First-order (discontinuous) phase transitions in deeper architectures ($D \ge 2$), signaled by coexisting minima and hysteresis in the loss landscape, with a discontinuity in the global minimizer at a critical parameter $\gamma^*$ (Ziyin et al., 2022).

The order parameter, such as $\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$0 for a matrix $\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$1 in a deep linear network, exhibits mean-field scaling near the transition, while the loss landscape exhibits a double-well structure and latent heat analogous to physical systems.

3. Phase Field–Inspired Losses for Implicit Neural Representations

In geometric deep learning, particularly implicit neural representations (INRs) for surface modeling, phase transition–inspired losses are constructed from phase-field models of two-phase fluids. The "PHASE" loss defines a scalar field $\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$2, whose zero level set represents the surface, and employs a Van-der-Waals–Cahn–Hilliard regularizer

$\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$3

with $\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$4, enforcing $\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$5 away from the interface. The limiting behavior as $\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$6 guarantees minimizers with proper binary occupancy, exact data fitting on the zero set, and minimal perimeter (surface area) among all feasible solutions—an inductive bias absent from purely local losses (e.g., cross-entropy or SDF regression). This structure unifies binary occupancy and distance regressor approaches, and delivers state-of-the-art performance in benchmark surface reconstruction tasks (Lipman, 2021).

Component	Formulation	Interpretation
Reconstruction term	$\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$7	Soft-mass constraint, surface fitting
WCH regularizer	$\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$8	Phase separation/minimal area
Normal/gradient loss (optional)	$\ell(w, a) = E_x[\ell_0(w, x)] + a R(w)$9 or $\ell_0$0	Enforce correct gradient/norm

Taking the $\ell_0$1 limit yields a binary indicator $\ell_0$2, with the corresponding interface minimizing perimeter and passing through the data, paralleling the minimal-surface property of two-phase systems.

4. Loss–Complexity Landscapes, Duality, and Susceptibility

The "loss–complexity landscape" framework introduces computable complexity proxies (Comp) and examines the trade-off between empirical loss and model complexity through Kolmogorov-style structure functions,

$\ell_0$3

and their convex Legendre dual, the free energy (action)

$\ell_0$4

Parameterizing by an inverse temperature $\ell_0$5, one constructs a family of $\ell_0$6-smoothed free energies,

$\ell_0$7

For large $\ell_0$8, these recover sharp transitions in the minimizer: as $\ell_0$9 passes through critical values, the set of optimal models jumps, and the complexity susceptibility $R(w)$0 exhibits pronounced peaks. Such peaks precisely signal trade-off phase transitions—elbow points in generalization/overfitting for tasks like regression or tree-depth selection (Kolpakov, 17 Jul 2025).

Empirical studies confirm that these transitions correspond to the optimal selection of model order, balancing fit and complexity, and the scaling of susceptibility enables algorithmic detection of the critical value.

5. Dynamical and Dissipation-Driven Phase Transition Losses

Phase transition–inspired losses also appear in non-equilibrium and dissipative quantum systems. In the non-Hermitian Kondo model, introducing two-body loss at a quantum impurity is mathematically equivalent to promoting the Kondo coupling to a complex parameter. The system is exactly solvable via a Bethe Ansatz, with the phase diagram dictated by a control angle $R(w)$1 and a generalized Kondo temperature $R(w)$2.

As $R(w)$3 increases, the system traces a path through a Kondo screened phase ($R(w)$4), a non-Hermitian Yu–Shiba–Rusinov analog phase ($R(w)$5), and an unscreened local moment phase ($R(w)$6). Critical points mark dynamical phase transitions associated with divergent time scales (lifetime and binding-energy scales), and observables such as impurity spin polarization and scattering phase shift provide rigorous phase diagnostics. The loss-induced breakdown of screening and critical slowing-down directly correspond to phase transitions in the non-equilibrium dynamics (Kattel et al., 2024).

6. Dissipation, Hysteresis, and Limits in First-Order Transitions

In solid-state transitions and caloric materials, dissipation losses and hysteresis exhibit phase transition structure that fundamentally limits performance. For first-order magnetostructural transitions (e.g., in Ni–Co–Mn–Ti Heusler alloys), the transition entropy change $R(w)$7 vanishes at a compensation temperature $R(w)$8, below which no transition can occur. As the system approaches $R(w)$9, hysteresis losses ($a$0) and susceptibility blow up, rendering refrigeration cycles ineffective at cryogenic temperatures. The sharp increase in dissipation at $a$1, universal across first-order caloric systems with non-negligible hysteresis, marks a dynamic phase boundary in the accessible operational regime for caloric cooling (Beckmann et al., 2023).

Material System	Phase Transition Signature	Loss/Operational Impact
Ni–Co–Mn–Ti Heusler	$a$2, $a$3	Hysteresis $a$4, cooling cycle fails
Other first-order calorics	Susceptibility peak, transformation arrest	Fundamental limit on refrigeration temperature

Mitigation strategies include tuning operating regimes away from $a$5, minimizing intrinsic hysteresis, and designing composite systems to avoid pronounced loop broadening.

7. Practical Implementations and Applications

Phase transition–inspired losses admit several implementation strategies adapted to target models and regimes:

• Training neural networks with Landau-polynomial regularizers exhibits tunable continuous/discontinuous changes in learned weights and generalization (Ziyin et al., 2022).
• Loss–complexity landscapes leverage algorithms such as simulated annealing and direct free energy minimization, guiding selection via susceptibility and critical-point diagnostics. Bayesian optimizers and ensemble-based approaches extend practical reach (Kolpakov, 17 Jul 2025).
• In geometric deep learning, the PHASE loss combined with signal embeddings (e.g., Fourier features) ensures high-resolution surface reconstruction while suppressing extraneous, nonminimal artifacts through global minimal-perimeter bias (Lipman, 2021).
• Material design and caloric cooling cycles are informed by explicit measurements and predictions of dissipation phase boundaries and susceptibility maxima, guiding selection of optimal compositional and operational parameters (Beckmann et al., 2023).

Adoption of such losses provides a principled path to global regularity, minimality, and robust control of abrupt changes in model complexity, enabling the transfer of universal phase-transition phenomena from physics to optimization and learning.