Data-Driven Quasi-Steady Model

• Data-driven quasi-steady models are learned representations that identify steady regimes while parameterizing departures from steady behavior in complex systems.
• They employ techniques like sparse regression, CNN surrogates, and state-space decompositions to derive interpretable governing equations and efficient reduced-order models.
• Applications span granular rheology, heat conduction, aeroelasticity, turbulence, and soil phosphorus, showcasing their versatility and practical impact.

A data-driven quasi-steady model is a learned representation of a physical or biological system in which the dominant behavior is expressed either as a steady-state mapping or as a low-order evolution law whose steady limit is explicit and whose departures from steady behavior can be parameterized from data. In recent work, the term spans several distinct constructions: a discovered ordinary differential equation for effective friction in sheared granular materials across steady and transient states; a convolutional surrogate for steady-state heat conduction; low-order aeroelastic and flapping-wing models that separate quasi-steady contributions from added-mass and unsteady terms; a regression-based correction of Reynolds stresses in a parallel plane quasi-steady turbulence flow case; and a soil-phosphorus model in which quasi-steady uptake emerges from multiscale chemical and biological dynamics (Han et al., 18 Sep 2025, Peng et al., 2020, Hickner et al., 2021, Zhu et al., 2020, Kamimizu et al., 26 Aug 2025, Moyles et al., 2019).

1. Meanings of “quasi-steady” in data-driven modeling

In the cited literature, “quasi-steady” does not denote a single formalism. In steady heat conduction, the term refers to the direct prediction of steady-state solutions for arbitrary geometries governed by Laplace’s equation, with no explicit time evolution in the learned model. In sheared granular rheology, quasi-steady behavior appears as the steady limit of a discovered differential law for the effective friction coefficient, while the same law also governs relaxation, hysteresis, and shear-reversal transients. In aeroelastic and flapping-wing problems, quasi-steady terms are treated as one physically interpretable component within a broader decomposition that also includes inertial or viscous-history effects. In soil-phosphorus mobility, quasi-steady uptake denotes long-lived plateaus generated by separation of time scales rather than strict stationarity (Peng et al., 2020, Han et al., 18 Sep 2025, Hickner et al., 2021, Kamimizu et al., 26 Aug 2025, Moyles et al., 2019).

A recurrent misconception is to equate quasi-steady with fully time-independent modeling. The granular and aeroelastic studies show the opposite structure: the quasi-steady contribution is identifiable only after separating it from transient dynamics, and the resulting model remains useful precisely because it retains those transient terms. A second misconception is to equate data-driven modeling with opaque black-box prediction. The granular PINNSR-DA framework, the aeroelastic state-space formulation, and the refined flapping-wing quasi-steady model are all explicitly organized around interpretable terms and coefficients rather than end-to-end prediction alone (Han et al., 18 Sep 2025, Hickner et al., 2021, Kamimizu et al., 26 Aug 2025).

2. Governing-equation discovery for sheared granular materials

The most explicit formulation of a data-driven quasi-steady model in the supplied material is the PINNSR-DA framework for sheared granular materials. PINNSR-DA combines physics-informed neural networks with sparse regression and machine-learning-based dimensional analysis to discover a parsimonious, dimensionally consistent governing equation from noisy discrete element method data. Applied to oscillatory shear flow, it yields an ordinary differential equation for the evolution of effective friction $\mu=\tau/P$:

$$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$

The three terms are interpreted respectively as linear response, energy dissipation, and nonlinear response. The same framework writes the coefficients in parameterized form,

$$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$

where the dominant dimensionless number is

$\chi=\frac{H}{d_p}\frac{E_p}{P}.$

The empirical coefficient laws are logarithmic, $f_i(\chi)=a_i\log_{10}(\chi)+b_i$, and the dimensionless relaxation time depends primarily on $\chi$, with larger $\chi$ corresponding to shorter relaxation times. In physical terms, larger $\chi$ means stiffer particles, thicker flows, or lower confining pressure; the paper states that relaxation is shorter for stiffer particles and thicker flow layers (Han et al., 18 Sep 2025).

The steady or quasi-static limit follows by setting $d\mu/dt=0$, which gives

$$C_1+C_2\mu\,\mathrm{sgn}\!\left(\frac{d\gamma}{dt}\right)+C_3\mu^2=0.$$

The resulting steady solution is independent of shear-rate magnitude, consistent with rate-independent quasi-static granular behavior. Under time-dependent loading, the same equation governs relaxation after shear reversal, hysteresis, and stress overshoots, and for canonical transients the equation admits an analytical Riccati-type solution. The paper further states that the discovered macroscopic terms correspond to meso-scale microstructure evolution through the fabric tensor and contact network anisotropy. This suggests a characteristic feature of data-driven quasi-steady modeling in rheology: steady constitutive structure is retained, but it is embedded in a transient closure rather than treated as a stand-alone law (Han et al., 18 Sep 2025).

3. Steady-state field surrogates in heat conduction

A different instantiation appears in data-driven modeling of geometry-adaptive steady heat transfer. Here the physical problem is steady-state heat conduction in two dimensions, governed by Laplace’s equation,

$$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$0

The learned object is not a reduced evolution equation but a direct map from geometry to temperature field. Geometry is encoded by a signed distance function,

$$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$1

which distinguishes points inside, on, and outside the boundary and preserves metric proximity information that a binary image does not. The network is a CNN encoder-decoder with input size $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$2, five convolutional layers, five deconvolution layers, ReLU activations except at the output, and a masked mean-squared-error loss with $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$3 regularization. The preferred loss ignores errors inside the hot object, and optimization uses Adam with mini-batch training and the parameters $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$4, $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$5, and $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$6 (Peng et al., 2020).

The training set consists of triangles, quadrilaterals, pentagons, hexagons, and dodecagons with random variations in size, shape, orientation, and location. Ground truth is generated with OpenFOAM and SnappyHexMesh. After training, the model predicts entire $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$7 temperature fields in $$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$8–$$\frac{d\mu}{dt}=C_1\frac{d\gamma}{dt}+C_2\mu\left|\frac{d\gamma}{dt}\right|+C_3\mu^2\frac{d\gamma}{dt}.$$9 seconds on a GPU, whereas the corresponding OpenFOAM simulations take approximately $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$0–$$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$1 seconds, yielding a speedup of three to four orders of magnitude. On the simple-shape test set, average relative errors are reported as $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$2–$$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$3 with SDF and the preferred loss; on unseen complex geometries such as “circle,” “locomotive,” and “human,” reported accuracies are $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$4, $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$5, and $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$6, with overall validation accuracy in the $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$7–$$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$8 range. The paper explicitly characterizes the result as a data-driven reduced-order model for quasi-steady or steady-state heat conduction (Peng et al., 2020).

This use of “quasi-steady” differs from the granular case. The model produces the steady solution directly, rather than integrating a transient law toward it. A plausible implication is that data-driven quasi-steady models divide into at least two categories: steady-state field surrogates, and transient-capable reduced laws whose steady limit remains analytically identifiable.

4. Separation of quasi-steady and unsteady forces in aerodynamic systems

In aeroelastic modeling for control, the data-driven quasi-steady component is embedded in a low-order linear state-space system identified from direct numerical simulations of flow past a flexible wing at low Reynolds number. The outputs are lift $$\frac{d\mu}{dt}=f_1(\chi)\frac{d\gamma}{dt}+f_2(\chi)\mu\left|\frac{d\gamma}{dt}\right|+f_3(\chi)\mu^2\frac{d\gamma}{dt},$$9 and a deformation measure $\chi=\frac{H}{d_p}\frac{E_p}{P}.$0, and the latent state $\chi=\frac{H}{d_p}\frac{E_p}{P}.$1 captures wake, viscous, and structural transients. The model writes the output as the sum of a latent contribution $\chi=\frac{H}{d_p}\frac{E_p}{P}.$2, a quasi-steady term proportional to instantaneous angle of attack, and added-mass terms proportional to pitch rate and acceleration. The quasi-steady lift slope is not fixed at the classical inviscid value but replaced by an empirical coefficient $\chi=\frac{H}{d_p}\frac{E_p}{P}.$3 extracted from steady-state data after transients decay. Added-mass coefficients $\chi=\frac{H}{d_p}\frac{E_p}{P}.$4 and $\chi=\frac{H}{d_p}\frac{E_p}{P}.$5 are identified from the impulse response, and the transient dynamics are identified with the Eigensystem Realization Algorithm using integrated Markov parameters and truncated singular value decomposition of Hankel matrices. The resulting reduced-order model is validated against aggressive maneuvers and then used in model predictive control while constraining maximum wing deformation (Hickner et al., 2021).

The flapping-wing study refines the quasi-steady model itself by discovering additional mechanisms from CFD data. Using a library of 5,548 candidate kinematic functions and SINDy with STRidge and recursive feature elimination, it identifies three mechanisms that had been qualitatively recognized but not formulated: the effect of advance ratio, the effect of spanwise kinematic velocity, and the rotational Wagner effect. The advance-ratio correction decomposes $\chi=\frac{H}{d_p}\frac{E_p}{P}.$6 into separate body, flapping, and coupling contributions with distinct force coefficients; the spanwise correction introduces spanwise blade-element terms and cross-terms; and the rotational Wagner effect introduces a lagged rotational-circulation contribution proportional to $\chi=\frac{H}{d_p}\frac{E_p}{P}.$7. Relative to the conventional quasi-steady model, the mean error in hawkmoth forward and hovering flight decreases from approximately $\chi=\frac{H}{d_p}\frac{E_p}{P}.$8 to $\chi=\frac{H}{d_p}\frac{E_p}{P}.$9–$f_i(\chi)=a_i\log_{10}(\chi)+b_i$0, and the error in fruit fly maneuvering flight decreases to $f_i(\chi)=a_i\log_{10}(\chi)+b_i$1–$f_i(\chi)=a_i\log_{10}(\chi)+b_i$2 (Kamimizu et al., 26 Aug 2025).

Taken together, these two studies establish a precise methodological distinction. In the aeroelastic case, the quasi-steady term is one identifiable component within a broader transient state-space closure. In the flapping-wing case, the quasi-steady model itself is reformulated by adding previously omitted mechanisms. This suggests that “data-driven quasi-steady model” may refer either to decomposition of forces into quasi-steady and non-quasi-steady parts, or to data-driven revision of the quasi-steady term set itself (Hickner et al., 2021, Kamimizu et al., 26 Aug 2025).

5. Regression correction in quasi-steady turbulence modeling

In turbulence modeling for thermal hydraulics, the quasi-steady setting is a fully developed turbulent flow between parallel plates. The data-driven model does not discover a new governing equation; instead, it learns a surrogate mapping from RANS-derived features to DNS Reynolds stresses:

$f_i(\chi)=a_i\log_{10}(\chi)+b_i$3

The high-fidelity targets are Reynolds stress tensor fields from direct numerical simulation, while the inputs are normalized strain rate, normalized rotation rate, and normalized pressure gradient extracted from baseline RANS simulations using standard $f_i(\chi)=a_i\log_{10}(\chi)+b_i$4–$f_i(\chi)=a_i\log_{10}(\chi)+b_i$5 with wall functions in OpenFOAM. Feature processing follows invariance ideas associated with Ling et al. (2016), and the regression model is Gaussian Process Regression implemented with the GPML Matlab toolbox, using the isotropic squared exponential kernel covSEiso and Gaussian likelihood likGauss. Training data are refined for even distribution to reduce bias toward densely sampled regions (Zhu et al., 2020).

Validation applies the learned stresses to a previously unseen Reynolds-number case and then re-solves the RANS equations for the mean velocity profile. The paper reports good agreement with DNS over most of the channel, especially in the log-law region, and notes that residuals are negligible above the wall-function region when DNS stresses are inserted directly. It also reports uncertainty bounds from the Gaussian process, with larger uncertainty where training coverage is sparse. No single-number error metric is reported; instead, assessment is based on velocity profiles, normalized Reynolds stress profiles, uncertainty bands, and residual plots (Zhu et al., 2020).

This example broadens the category further. The learned object is neither a stand-alone quasi-steady constitutive law nor a direct steady-state field map. Rather, it is a correction operator for a lower-fidelity quasi-steady solver. A plausible implication is that data-driven quasi-steady modeling often functions as an intermediate layer between first-principles closure and high-fidelity reference data.

6. Quasi-steady uptake and time-scale separation in soil phosphorus mobility

The soil-phosphorus study is mathematically different from the machine-learning-focused examples but is central to the broader meaning of quasi-steady dynamics. The model tracks free orthophosphate $f_i(\chi)=a_i\log_{10}(\chi)+b_i$6, adsorbed inorganic phosphorus $f_i(\chi)=a_i\log_{10}(\chi)+b_i$7, plant biomass phosphorus $f_i(\chi)=a_i\log_{10}(\chi)+b_i$8, copiotroph biomass phosphorus $f_i(\chi)=a_i\log_{10}(\chi)+b_i$9, and oligotroph biomass phosphorus $\chi$0, with adsorption, desorption, fertilization, Monod-type uptake, and biomass loss. After non-dimensionalization, the reduced system is

$\chi$1

The critical parameter is

$\chi$2

the ratio of chemical adsorption to copiotroph death. Large $\chi$3 corresponds to fast chemistry and slow biomass turnover, producing rapid equilibration of nutrient pools and slow community change; small $\chi$4 allows oscillatory behavior. Four equilibrium states emerge: oligotroph-only, copiotroph-only, plant-only, and coexistence of all species (Moyles et al., 2019).

The paper shows that quasi-steady uptake can coexist with microbial community reassembly. Soil and plant phosphorus content can become unresponsive to increased fertilization, while the additional fertilization supports the copiotrophs and shifts community composition. It therefore emphasizes the importance of time-series measurements, since endpoint measurements can mistake a long-lived quasi-steady plateau for a definitive equilibrium. Relative to the other studies, this is a quasi-steady model in the classical dynamical-systems sense: the quasi-steady state is an emergent regime of a mechanistic ODE system rather than a learned surrogate (Moyles et al., 2019).

7. Cross-cutting methodological characteristics and limitations

Across these studies, high-fidelity or mechanistic references are indispensable: DEM in granular shear, OpenFOAM solutions of Laplace’s equation in heat conduction, DNS in aeroelasticity and turbulence, CFD in flapping-wing aerodynamics, and an explicit multiscale uptake model in soil phosphorus. The learned or reduced object varies markedly: a sparse ODE with dimensionless coefficient laws, a CNN geometry-to-field map, a linear state-space model with interpretable force partitions, a Gaussian-process closure correction, and a mechanistic slow-fast system with quasi-steady plateaus. This suggests that “data-driven quasi-steady model” is best understood as a family of constructions rather than a single architecture (Han et al., 18 Sep 2025, Peng et al., 2020, Hickner et al., 2021, Zhu et al., 2020, Kamimizu et al., 26 Aug 2025, Moyles et al., 2019).

Several limitations also recur. The aeroelastic framework assumes linear response and may break down near stall or under large deformations. The turbulence study shows sensitivity to training-set density, especially away from the wall-function region. The heat-conduction surrogate is trained on a specific two-dimensional problem class and its error increases when objects move away from the training distribution. The granular formulation is derived from oscillatory shear DEM data and then parameterized by a dominant dimensionless number, which supports extrapolation to untrained stiffnesses or system sizes but remains tied to the discovered coefficient scaling. The flapping-wing study improves quasi-steady accuracy substantially, yet it does so by augmenting the mechanism set rather than eliminating the underlying trade-off between interpretability and full high-fidelity fidelity. In all cases, the literature treats quasi-steady modeling not as a replacement for physics, but as a route to physically interpretable, computationally efficient approximations whose validity depends on the data regime and on explicit assumptions about time scales, geometry, or operating conditions (Han et al., 18 Sep 2025, Peng et al., 2020, Hickner et al., 2021, Zhu et al., 2020, Kamimizu et al., 26 Aug 2025, Moyles et al., 2019).