Junning Jiao

Porous electrodes couple electrical, chemical, mechanical, hydraulic, and thermal fields, yet conventional frequency-domain diagnostics interrogate only one of them: electrochemical impedance spectroscopy (EIS) the electrical response and dynamic mechanical analysis (DMA) the mechanical. Each reads a diagonal entry of the multiphysical constitutive matrix; the off-diagonal couplings—the basis of cross-domain transduction, and of degradation in electrodes—enter only implicitly. Starting from linear irreversible thermodynamics, we formulate a general theory of multiphysical impedance spectroscopy, in which perturbing one field and measuring the conjugate response of another probes off-diagonal entries explicitly. Specializing to the electro-chemo-mechanical pathway yields a closed-form theory of mechano-electrochemical impedance spectroscopy (MEIS), in which a small harmonic current is applied and the stack stress is measured. The impedance factorizes into a chemical-accumulation term multiplying the sum of a chemo-mechanical and a poro-mechanical kernel, with a porosity-accommodation bridge function derived from a Helmholtz free energy and a three-phase closure that interpolates between unsaturated and Biot-saturated limits. Because the mechanical kernel reduces to an equivalent spring—dashpot network, measured spectra can be fitted as in EIS or DMA with each element retaining a definite physical meaning, and MEIS emerges as one member of a broader family of cross-coupled spectroscopies.

Phase-field models are widely used to describe phase transitions and interface evolution in various scientific disciplines. In this Tutorial, we present two neural network methods for solving them. The first method is based on physics-informed neural networks (PINNs), which enforce the governing equations and boundary/initial conditions in the loss function. The second method is based on deep operator neural networks (DeepONets), which treat the neural network as an operator that maps the current state of the field variable to the next state. Both methods are demonstrated with the Allen–Cahn equation in one dimension, and the results are compared with the ground truth. This Tutorial also discusses the advantages and limitations of each method, as well as the potential extensions and improvements.