KA-CRNN: Interpretable Kinetic Modeling

• KA-CRNN is a framework that generalizes chemical reaction neural networks by parameterizing kinetic parameters as continuous functions of external variables like SOC or pressure using univariate Kolmogorov–Arnold activations.
• It integrates a CRNN core with KAN layers to capture SOC- or pressure-dependent kinetic behaviors while strictly enforcing Arrhenius and mass action laws.
• Demonstrated applications in Li-ion cathode decomposition and pressure-dependent kinetics show high model fidelity with low mean squared error and enhanced interpretability.

The Kolmogorov–Arnold Chemical Reaction Neural Network (KA-CRNN) framework generalizes classical Chemical Reaction Neural Networks by representing each kinetic parameter as a continuous function of an external variable—typically state of charge (SOC) or pressure—using univariate Kolmogorov–Arnold Network (KAN) activations. This approach enables fully interpretable, assumption-free inference of extended kinetic behaviors (such as SOC- or pressure-dependent rates and enthalpies) from data, while preserving strict adherence to Arrhenius and mass action laws. KA-CRNNs have been demonstrated in the context of learning SOC-dependent thermal decomposition kinetics for Li-ion cathodes and pressure-dependent elementary chemical kinetics, providing accurate, physically constrained, and highly interpretable kinetic models (Koenig et al., 17 Dec 2025, Koenig et al., 10 Nov 2025).

1. Theoretical Foundations

1.1. Kolmogorov–Arnold Representation Theorem

The Kolmogorov–Arnold representation theorem states that any continuous multivariate function $f: [0,1]^m \to \mathbb{R}$ can be decomposed as

$$f(x_1,\dots,x_m) = \sum_{q=0}^{2m} \Phi_q\left(\sum_{p=1}^m \phi_{q,p}(x_p)\right)$$

with $\Phi_q$ and $\phi_{q,p}$ continuous univariate functions. For KA-CRNNs, this principle motivates a neural architecture in which each kinetic parameter—such as pre-exponential factor $A$, activation energy $E_a$, or enthalpy change $\Delta H$—is represented as a learnable continuous function of a single external variable (e.g., SOC or pressure) via a univariate KAN layer.

1.2. Motivation and Distinction from Standard CRNNs

Standard CRNNs fit scalar kinetic parameters (fixed in the external variable), so they cannot represent continuous dependence on SOC, pressure, or other coordinates. In KA-CRNNs, each parameter (e.g., $A_{\text{R}_i}(z)$, $E_{a,\text{R}_i}(z)$ with $z$ denoting SOC or $P$) is parameterized as a continuous function using learnable basis expansions, enabling the model to mirror gradual or abrupt kinetic regime changes. This yields an interpretable, continuous mapping from operational variables to mechanistic rates (Koenig et al., 17 Dec 2025, Koenig et al., 10 Nov 2025).

2. Network Architecture and Chemical Encoding

2.1. Structural Overview

Input	Hidden Layers	Output
External variable (SOC/Pressure)	Bank of KAN activations for each kinetic parameter	ODE-integrated species trajectories
Temperature profile	Arrhenius/mass-action feed-forward CRNN core	Model heat release or concentration
Species initial mass fractions

In the thermal decomposition (SOC-dependent) setting (Koenig et al., 17 Dec 2025):

• Input: normalized SOC $z \in [0,1]$, imposed $T(t)$, latent $c_i(t)$.
• Hidden: for each scalar kinetic parameter $p_i$, a Chebyshev-polynomial KAN computes $p_i(z)$; the CRNN core computes $r_i = f(p_i(z), c_i, T)$.
• Output: species mass fraction trajectories and heat release rate $\dot Q(t) = \sum_i \Delta H_i(z) r_i$.

Mechanistically, the reaction network is embedded in the architecture:

• R1: $C_\ell \to C_s$ (“layered $\to$ spinel”); R2: $C_s \to C_r + \nu\,O_2$ (“spinel $\to$ rock-salt + O$_2$”); R3: $O_2 + C_e \to C_{\text{prod}}$ (O$_2$-driven electrolyte oxidation).
• R1 and R2: SOC-dependent kinetics; R3: fixed parameters, rate proportional to O$_2$ produced in R2.

For pressure-dependent reactions (Koenig et al., 10 Nov 2025), the architecture is analogous, with normalized $P$ as input, KAN layers for each parameter, and the reaction network ODE integrated accordingly.

2.2. Physical Encoding and Interpretability

Mechanistic reaction topologies are strictly enforced by the architectural design. Each kinetic parameter has a single interpretable univariate mapping (e.g., $A_i(z)$, $E_{a,i}(z)$), which can be visualized or symbolically regressed post hoc.

3. Mathematical Framework

3.1. Rate Law and ODE System

Each elemental reaction $i$ is governed by:

$$r_i = A_i(z)\,T^{b_i(z)}\,c_i^{n_i(z)}\,\exp\left[-\frac{E_{a,i}(z)}{RT}\right]$$

where $z$ is SOC or pressure as dictated by the application. In log space:

$$\ln r_i = n_i \ln c_i + \ln A_i + b_i \ln T - \frac{E_{a,i}}{RT}$$

For the SOC-dependent Li-ion cathode decomposition:

• $c_1$, $c_2$, $c_3$ correspond to layered, spinel, and O$_2$ mass fractions, respectively.
• The ODE system, \begin{align*} \frac{dc_1}{dt} &= -r_1 \ \frac{dc_2}{dt} &= +r_1 - r_2 \ \frac{dc_3}{dt} &= +\nu(z) r_2 - r_3 \end{align*}
• Total heat release:

$\dot Q(t) = \sum_{i=1}^3 \Delta H_i(z) r_i(t)$

3.2. KAN Basis Expansion for Kinetic Parameters

Each SOC- or pressure-dependent parameter $p_i$ is parameterized as:

$$p_i(z) = \sum_{n=0}^N w_{i,n} \psi_n(z), \quad \psi_n(z) = \cos(n \arccos(z))$$

In practice, $N=10$ is typical for SOC or pressure; the set $\{w_{i,n}\}$ is learned to fit observed data.

3.3. Loss Function and Regularization

The loss,

$$\mathcal{L} = \frac{1}{K} \sum_{k=1}^K \int (\dot Q_k^{\text{data}}(T) - \dot Q_k^{\text{model}}(T))^2\,dT + \mathcal{L}_{\text{mono}} + \mathcal{L}_{\text{min}} + \mathcal{L}_{\text{max}}$$

includes:

• $\mathcal{L}_{\text{mono}}$, which penalizes non-monotonicity in key parameters (e.g., ensures $\Delta H_2(z)$ and $\nu(z)$ are non-decreasing in SOC),
• $\mathcal{L}_{\text{min}}$, which enforces $n_i \geq 1$,
• $\mathcal{L}_{\text{max}}$, which prevents pathological parameter values.

This structure is analogous when training on pressure-dependent data, with terms modified for the relevant physical and kinetic constraints.

4. Training Methodology

4.1. Data Preprocessing and Sampling

For SOC-dependent kinetics, experimental DSC scans are collected at multiple linearly spaced SOCs per cathode chemistry; one scan is held out for testing. Heat release is computed as $\dot Q(T)$, and SOC is normalized to $[0,1]$. For pressure dependence, training sets consist of trajectory data over a grid of temperatures and normalized pressures.

4.2. Neural ODE Integration and Optimization

The network integrates the chemical reaction ODEs (species and heat) using Neural-ODE libraries (e.g., DifferentialEquations.jl). Gradients for all model parameters (KAN weights, initial conditions, etc.) are computed via forward-mode automatic differentiation (e.g., ForwardDiff.jl).

Optimization employs the ADAM algorithm, typically with step size $5 \times 10^{-4}$; training proceeds until plateau, generally $10^3$–$10^4$ steps. Physics-informed penalties are incorporated to avoid overfitting and to regularize model extrapolation.

4.3. Regularization and Physicality

Regularizers enforce physically plausible monotonicity, parameter range, kinetic order constraints, and smoothness in the KAN representations. For pressure-dependent cases, initialization at the known high-pressure (or high-SOC) limiting parameter values expedites convergence.

5. Interpretability and Model Validation

5.1. Parameter Variation with External Variables

Learned parameter functions capture known or hypothesized physical transitions:

• In Li-ion cathode decomposition (Koenig et al., 17 Dec 2025), $A_2(z)$, $E_{a,2}(z)$, and the O$_2$ stoichiometric jump sharply at a “critical SOC,” recapitulating the lattice oxygen release and phase transformation reported in literature.
• The functional dependence is continuous, interpretable, and can be visualized directly.
• In pressure-dependent kinetics (Koenig et al., 10 Nov 2025), learned KAN curves reproduce expected falloff behavior and are directly human-inspectable.

5.2. Mechanistic and Domain Insight

Interpretability is enforced at two levels: (1) strict physical/chemical network encoding in the CRNN ODE, (2) mapping each parameter to an explicitly learnable function of the external variable via a sparse basis. Sudden regime transitions in kinetic parameters align with domain knowledge (e.g., abrupt O$_2$ release at high SOC, pressure falloff in combustion). Symbolic regression can extract closed-form kinetic laws from learned KANs.

5.3. Generalization Fidelity

KA-CRNNs reproduce all training curves with low mean squared error and correctly interpolate to held-out external variable values (e.g., mid-range SOC or pressure). The continuous basis eliminates artifacts at intermediate values and outperforms piecewise or discrete fits. Quantitative metrics on test sets (e.g., $R^2 > 0.98$ for thermal peak properties) demonstrate model fidelity.

6. Extensions, Limitations, and Applications

6.1. Framework Generality

KA-CRNNs can be immediately extended to:

• Additional external coordinates (multivariate KANs for $T$, $P$, SOC, etc.),
• More complex reaction networks (a KAN per parameter per step, or sharing among reactions),
• Symbolic extraction of interpretable kinetic rate laws from learned representations.

6.2. Limitations and Recommendations

Current deployments focus on univariate external variable dependencies; multidimensional KAN layers would extend capability but increase data and computational demands. For sparse data (Koenig et al., 10 Nov 2025), incorporating analytic priors or hierarchical approaches may be essential. In stiff or highly multi-scale ODE systems, advanced solvers or architectural regularizations are advisable.

6.3. Application Contexts

Recommended applications include:

• Thermal decomposition and abuse prediction for Li-ion batteries across all SOCs (Koenig et al., 17 Dec 2025),
• Pressure- or concentration-dependent kinetics in combustion, catalysis, atmospheric chemistry,
• Enzymatic and materials science scenarios with field-coupled kinetics (Koenig et al., 10 Nov 2025).

7. Comparative and Methodological Significance

KA-CRNNs offer a domain-informed, physics-consistent approach for inferring complex, continuous kinetic parameter dependencies directly from data. They outperform black-box neural networks and standard interpolative models (e.g., PLOG) both in accuracy and interpretability. All kinetic dependencies remain physically explicit and subject to posterior analysis. This suggests KA-CRNNs are a broadly applicable tool for kinetic inference when external variable dependence is critical and interpretability essential (Koenig et al., 17 Dec 2025, Koenig et al., 10 Nov 2025).