.. _model-fractional-maxwell-liquid: Fractional Maxwell Liquid (Fractional) ====================================== Quick Reference --------------- - **Use when:** Viscoelastic liquid, power-law relaxation without terminal flow plateau - **Parameters:** 3 (:math:`G_m`, :math:`\alpha`, :math:`\tau_\alpha`) - **Key equation:** :math:`G(t) = G_m t^{-\alpha} E_{1-\alpha,1-\alpha}(-t^{1-\alpha}/\tau_\alpha)` - **Test modes:** Oscillation, relaxation, creep, flow curve - **Material examples:** Polymer melts (linear/branched), concentrated polymer solutions, complex fluids .. include:: /_includes/fractional_seealso.rst Overview -------- The Fractional Maxwell Liquid (FML) model consists of a Hookean spring in series with a SpringPot element. This configuration describes materials with instantaneous elastic response at short times followed by power-law relaxation at intermediate to long times. The model is particularly effective for characterizing polymer melts, concentrated polymer solutions, and other viscoelastic liquids that exhibit both elastic memory and power-law relaxation without terminal flow. Unlike the Fractional Maxwell Gel which includes a dashpot for terminal flow, the FML model maintains power-law behavior across all time scales, making it ideal for materials that show persistent viscoelastic behavior without approaching pure viscous flow. Notation Guide -------------- .. list-table:: :widths: 15 40 20 :header-rows: 1 * - Symbol - Description - Units * - :math:`G_m` - Maxwell modulus (short-time elastic stiffness) - Pa * - :math:`\alpha` - Fractional order (0 < :math:`\alpha` < 1, controls relaxation spectrum breadth) - — * - :math:`\tau_\alpha` - Characteristic relaxation time - s\ :math:`^{\alpha}` * - :math:`E_{\alpha,\beta}(z)` - Two-parameter Mittag-Leffler function - — * - :math:`G^*(ω)` - Complex modulus - Pa * - :math:`G'(ω)` - Storage modulus (elastic component) - Pa * - :math:`G''(ω)` - Loss modulus (viscous component) - Pa * - :math:`J(t)` - Creep compliance - Pa\ :sup:`-1` * - :math:`\omega` - Angular frequency - rad/s * - :math:`t` - Time - s Physical Foundations -------------------- The FML model represents **viscoelastic liquids** with zero equilibrium modulus (Ge = 0), meaning the material flows under sustained stress. The physical structure consists of: 1. **Hookean spring (Gm)**: Provides instantaneous elastic response at short times. Represents chain/network stretching before relaxation mechanisms activate. 2. **SpringPot element**: Governs the relaxation dynamics through power-law viscoelasticity. The fractional order :math:`\alpha` controls the breadth of the relaxation spectrum. The series configuration ensures that sustained stress eventually leads to unbounded strain growth (flow), distinguishing this from solid-like models. **For FML specifically**, the fractional order :math:`\alpha` directly controls the slope in log-log plots of :math:`G'(\omega)` and :math:`G''(\omega)`, with both moduli scaling as :math:`\omega^{\alpha}` in the power-law region. Typical :math:`\alpha` ranges for FML applications: - Polymer melts (linear homopolymers): :math:`\alpha` ≈ 0.7-0.9 - Polymer melts (branched): :math:`\alpha` ≈ 0.5-0.7 - Concentrated polymer solutions: :math:`\alpha` ≈ 0.5-0.8 - Complex fluids (colloidal dispersions): :math:`\alpha` ≈ 0.4-0.7 Mathematical Foundations ------------------------ Mittag-Leffler Functions ~~~~~~~~~~~~~~~~~~~~~~~~ The FML model relies on the **two-parameter Mittag-Leffler function**: .. math:: E_{\alpha,\beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)} where :math:`\Gamma` is the gamma function. This generalization of the exponential function is essential for fractional viscoelasticity. **Key Properties:** - :math:`E_1,_1(z)` = exp(z) (recovers classical exponential) - :math:`E_{\alpha,\alpha(-t^\alpha)}` provides the characteristic power-law relaxation - Smoothly interpolates between short-time power-law and long-time stretched exponential Power-Law Relaxation Derivation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For the FML model, the relaxation modulus exhibits: .. math:: G(t) \sim t^{-\alpha} \quad \text{for } t \gg \tau_\alpha This power-law decay arises from the SpringPot constitutive equation and contrasts sharply with exponential decay (classical Maxwell). The power-law reflects a continuous distribution of relaxation times spanning many decades. Governing Equations ------------------- **Relaxation Modulus**: .. math:: G(t) = G_m t^{-\alpha} E_{1-\alpha,1-\alpha}\left(-\frac{t^{1-\alpha}}{\tau_\alpha}\right) where :math:`G_m` is the Maxwell modulus, :math:`E_{\alpha,\beta}(z)` is the two-parameter Mittag-Leffler function, and :math:`\tau_\alpha` is the characteristic relaxation time with units of s\ :sup:`alpha`. **Physical interpretation:** - Short times (:math:`t \ll \tau_\alpha`): :math:`G(t) \approx G_m` (elastic plateau) - Intermediate times: :math:`G(t) \sim t^{-\alpha}` (power-law relaxation) - Long times: Stretched exponential decay toward zero **Complex Modulus**: .. math:: G^*(\omega) = G_m \frac{(i\omega\tau_\alpha)^\alpha}{1 + (i\omega\tau_\alpha)^\alpha} Decomposing into storage and loss moduli: .. math:: G'(\omega) = G_m \frac{(\omega\tau_\alpha)^\alpha [1 + (\omega\tau_\alpha)^\alpha \cos(\alpha\pi/2)]}{1 + 2(\omega\tau_\alpha)^\alpha \cos(\alpha\pi/2) + (\omega\tau_\alpha)^{2\alpha}} .. math:: G''(\omega) = G_m \frac{(\omega\tau_\alpha)^\alpha \sin(\alpha\pi/2)}{1 + 2(\omega\tau_\alpha)^\alpha \cos(\alpha\pi/2) + (\omega\tau_\alpha)^{2\alpha}} **Frequency-Domain Behavior:** - High :math:`\omega` (:math:`\omega \gg 1/\tau_\alpha`): :math:`G' \to G_m`, :math:`G'' \to 0` (elastic plateau) - Intermediate :math:`\omega` (:math:`\omega \sim 1/\tau_\alpha`): :math:`G', G'' \sim \omega^\alpha` (power-law scaling, parallel slopes) - Low :math:`\omega` (:math:`\omega \ll 1/\tau_\alpha`): :math:`G' \sim \omega^{2\alpha}`, :math:`G'' \sim \omega^\alpha` (liquid-like terminal regime) **Creep Compliance**: .. math:: J(t) = \frac{1}{G_m} + \frac{t^\alpha}{G_m \tau_\alpha^\alpha} E_{\alpha,1+\alpha}\left(-\left(\frac{t}{\tau_\alpha}\right)^\alpha\right) **Physical interpretation:** - Short times: :math:`J(t) \approx 1/G_m` (elastic compliance) - Long times: Unbounded growth :math:`J(t) \to \infty` (liquid-like flow) The Mittag-Leffler function provides a smooth interpolation between exponential decay (when alpha=1) and stretched exponential or power-law relaxation (when 0 < alpha < 1): .. math:: E_{\alpha,\beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)} Parameters ---------- The Fractional Maxwell Liquid model has three parameters: .. list-table:: Parameters :header-rows: 1 :widths: 18 12 12 18 40 * - Name - Symbol - Units - Bounds - Notes * - ``Gm`` - :math:`G_m` - Pa - [1e-3, 1e9] - Maxwell modulus (short-time elasticity) * - ``alpha`` - :math:`\alpha` - dimensionless - [0, 1] - Fractional order (spectrum breadth) * - ``tau_alpha`` - :math:`\tau_\alpha` - s^alpha - [1e-6, 1e6] - Characteristic relaxation time **Parameter Interpretation:** - **Gm**: Instantaneous modulus reflecting chain/network stiffness. For polymer melts, relates to entanglement density via Gm ≈ :math:`G_N^0` (plateau modulus). Typical values: :math:`10^3-10^6` Pa for polymer melts. - **alpha**: Quantifies relaxation spectrum breadth. Lower :math:`\alpha` → broader spectra from molecular weight polydispersity, branching, or complex intermolecular interactions. For linear polymers, :math:`\alpha` ≈ 0.7-0.9; for branched polymers, :math:`\alpha` ≈ 0.5-0.7. - **tau_alpha**: Average relaxation time scale. Has unusual units (s\ :math:`^{\alpha}`) due to fractional calculus. For polymer melts, relates to molecular weight via :math:`\tau_\alpha \sim M_w^{3.4}`. Typical values: :math:`10 \times 10^{-3-10^3}` s depending on molecular weight and temperature. Validity and Assumptions ------------------------ - Linear viscoelastic assumption: strain amplitudes remain small (:math:`\gamma_0` < 5-10% typically). - Isothermal conditions: constant temperature throughout experiment. - Time-invariant material parameters: no aging, polymerization, or degradation. - Supported RheoJAX test modes: relaxation, creep, oscillation. - Fractional orders stay within (0, 1) to keep kernels causal and bounded. - Assumes liquid-like behavior: zero equilibrium modulus (Ge = 0), material flows under stress. What You Can Learn ------------------ This section explains how to translate fitted FML parameters into material insights and actionable knowledge. Parameter Interpretation ~~~~~~~~~~~~~~~~~~~~~~~~ **Fractional Order (** :math:`\alpha` **)**: The fractional order reveals molecular architecture and relaxation dynamics: - **0.7 <** :math:`\alpha` **< 0.9**: Narrow relaxation spectrum. Typical for linear, monodisperse polymer melts with well-defined entanglement dynamics. - **0.5 <** :math:`\alpha` **< 0.7**: Moderate spectrum breadth. Common in branched polymers, polydisperse melts, or concentrated solutions where multiple relaxation mechanisms coexist. - :math:`\alpha` **< 0.5**: Very broad spectrum. Indicates complex hierarchical relaxation (star polymers, H-polymers) or strong polydispersity. *For graduate students*: The fractional order connects to molecular weight distribution. For polymers, :math:`\alpha` ≈ 1/(1 + PDI/3) approximately, where PDI is the polydispersity index. Branching lowers :math:`\alpha` due to arm retraction and branch point hopping mechanisms. *For practitioners*: Use :math:`\alpha` to assess batch-to-batch consistency. A sudden drop in :math:`\alpha` suggests contamination with branched species or broadening of MWD. **Maxwell Modulus (Gm)**: The modulus reveals network/entanglement density: - **Gm ≈** :math:`G_N^0` **(plateau modulus)**: For entangled polymer melts, Gm should match the rubbery plateau from literature. Significant deviation suggests incomplete entanglement or dilution. - **Relationship to Me**: :math:`G_m = \rho R T / M_e` where Me is entanglement molecular weight. *For practitioners*: Track Gm as a QC metric. For polymer melts, Gm should be stable (±10%) across batches of the same grade. **Relaxation Time (** :math:`\tau_\alpha` **)**: The characteristic time connects to molecular weight: - **Scaling**: For linear polymers, :math:`\tau_\alpha \propto M_w^{3.4}` (reptation theory). - **Temperature dependence**: Follows WLF or Arrhenius behavior. *For practitioners*: Compare :math:`\tau_\alpha` to process timescales. For extrusion, ensure :math:`\tau_\alpha < 1/\dot{\gamma}_{process}` for complete relaxation. Material Classification ~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: FML Material Classification :header-rows: 1 :widths: 20 20 30 30 * - :math:`\alpha` Range - Spectrum Type - Typical Materials - Implications * - 0.8 < :math:`\alpha` < 1.0 - Very narrow - Monodisperse linear polymers - Near-Maxwellian, consider classical model * - 0.6 < :math:`\alpha` < 0.8 - Narrow-moderate - Commercial polymer melts - Standard processing behavior * - 0.4 < :math:`\alpha` < 0.6 - Broad - Branched polymers, blends - Complex flow behavior, longer relaxation * - :math:`\alpha` < 0.4 - Very broad - Highly branched, filled systems - Multiple mechanisms, difficult to process Diagnostic Indicators ~~~~~~~~~~~~~~~~~~~~~ Warning signs in fitted parameters: - :math:`\alpha` **→ 1**: Material is nearly Maxwellian. Consider using classical Maxwell for simpler interpretation and faster computation. - **Gm ≠** :math:`G_N^0`: If Gm differs significantly from tabulated plateau modulus, check for dilution, incomplete entanglement, or fitting errors. - :math:`\tau_\alpha` **inconsistent with Mw**: Compare to literature correlations. Large deviations suggest degradation or contamination. - **Poor fit at low frequencies**: Terminal behavior may not match FML predictions. Consider FMG (with dashpot) for materials showing true terminal flow. Application Examples ~~~~~~~~~~~~~~~~~~~~ **Polymer Grade Verification**: Fit FML to frequency sweep, compare :math:`\alpha` and :math:`\tau_\alpha` to specifications. A batch with lower :math:`\alpha` likely has broader MWD or unexpected branching. **Processing Optimization**: Use :math:`\tau_\alpha` to set residence times. For complete stress relaxation, ensure process time > :math:`5\tau_\alpha`. **Blend Analysis**: Lower :math:`\alpha` in blends indicates poor miscibility (separate relaxation modes) or broad combined MWD. Fitting Guidance ---------------- **Recommended Data Collection:** 1. **Frequency sweep** (SAOS): 3-5 decades (e.g., 0.01-100 rad/s) 2. **Test amplitude**: Within LVR (typically 0.5-5% strain) 3. **Coverage**: Ensure both elastic plateau and power-law regimes captured 4. **Temperature control**: ±0.1°C for polymer melts **Initialization Strategy:** .. code-block:: python # From frequency sweep |G*|(ω) Gm_init = high_freq_plateau # Elastic plateau tau_alpha_init = 1 / (frequency at steepest slope) alpha_init = slope in power-law region # Smart initialization (automatic in RheoJAX v0.2.0+) # Applied automatically when test_mode='oscillation' **Optimization Tips:** - Fit simultaneously to :math:`G'` and :math:`G''` for better constraint - Use log-weighted least squares - Verify power-law region (parallel :math:`G'`, :math:`G''` slopes) - Check residuals for systematic deviations **Common Pitfalls:** - **Insufficient high-frequency data**: Cannot determine Gm accurately - **Missing power-law regime**: Need broader frequency coverage - :math:`\alpha` **near 1**: Use classical Maxwell for simpler interpretation Usage ----- .. code-block:: python from rheojax.models import FractionalMaxwellLiquid from rheojax.core.data import RheoData import numpy as np # Create model instance model = FractionalMaxwellLiquid() # Fit to experimental data (smart initialization automatic) omega_exp = np.logspace(-2, 2, 50) G_star_exp = load_experimental_data() # Complex modulus model.fit(omega_exp, G_star_exp, test_mode='oscillation') # Inspect fitted parameters print(f"Gm = {model.parameters.get_value('Gm'):.2e} Pa") print(f"α = {model.parameters.get_value('alpha'):.4f}") print(f"τ_α = {model.parameters.get_value('tau_alpha'):.2e} s^α") # Predict relaxation modulus t = np.logspace(-3, 3, 100) data = RheoData(x=t, y=np.zeros_like(t), domain='time') data.metadata['test_mode'] = 'relaxation' G_t = model.predict(data) # Bayesian uncertainty quantification result = model.fit_bayesian( omega_exp, G_star_exp, num_warmup=1000, num_samples=2000, test_mode='oscillation' ) intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95) For more details, see :doc:`API reference `. Regimes and Behavior -------------------- The Fractional Maxwell Liquid exhibits characteristic behavior across different regimes: **Short-Time / High-Frequency Regime** (:math:`t \ll \tau_\alpha` or :math:`\omega \gg 1/\tau_\alpha`): The spring dominates, yielding purely elastic behavior: .. math:: G(t) \sim G_m, \quad G^*(\omega) \sim G_m The material behaves as an elastic solid with modulus :math:`G_m`. This regime captures the instantaneous response before relaxation mechanisms activate. **Intermediate Regime** (:math:`t \sim \tau_\alpha` or :math:`\omega \sim 1/\tau_\alpha`): The Mittag-Leffler function provides a smooth crossover between elastic plateau and power-law relaxation. This is the **fingerprint** of fractional viscoelasticity: .. math:: G'(\omega), G''(\omega) \sim \omega^\alpha \quad \text{(parallel slopes in log-log plot)} The loss tangent :math:`\tan\delta = G''/G'` exhibits a maximum at the characteristic frequency :math:`\omega \sim 1/\tau_\alpha`. **Long-Time / Low-Frequency Regime** (:math:`t \gg \tau_\alpha` or :math:`\omega \ll 1/\tau_\alpha`): The SpringPot controls the response with power-law behavior: .. math:: G(t) \sim G_m \left(\frac{t}{\tau_\alpha}\right)^{-\alpha}, \quad G^*(\omega) \sim G_m (i\omega\tau_\alpha)^\alpha For very low frequencies, terminal liquid-like behavior emerges: .. math:: G'(\omega) \sim \omega^{2\alpha}, \quad G''(\omega) \sim \omega^\alpha \quad \text{(G" > G')} Comparison with Classical Maxwell ---------------------------------- **Classical Maxwell (** :math:`\alpha` **= 1):** - Single relaxation time :math:`\tau` - Exponential relaxation: :math:`G(t) = G_m \exp(-t/\tau)` - Narrow relaxation spectrum (Lorentzian) - Low-frequency behavior: :math:`G' \sim \omega^2`, :math:`G'' \sim \omega` (classical liquid) **Fractional Maxwell Liquid (0 <** :math:`\alpha` **< 1):** - Continuous distribution of relaxation times - Power-law relaxation: :math:`G(t) \sim t^{-\alpha}` - Broad relaxation spectrum - Low-frequency behavior: :math:`G' \sim \omega^{2\alpha}`, :math:`G'' \sim \omega^\alpha` (generalized liquid) **When to Use Fractional:** - Power-law relaxation observed in stress relaxation experiments - Log-log plots of :math:`G'` and :math:`G''` show parallel slopes over multiple decades - Polymer melts with broad molecular weight distribution - Concentrated solutions with complex intermolecular interactions **When Classical Suffices:** - Single dominant relaxation time (linear homopolymers, dilute solutions) - Data span < 2 decades in frequency - Exponential decay observed experimentally Limiting Behavior ----------------- The FML model connects to classical models in limiting cases: - **alpha -> 1**: Recovers the classical Maxwell model with exponential relaxation: :math:`G(t) = G_m e^{-t/\tau_\alpha}` - **alpha -> 0**: Approaches purely elastic solid behavior: :math:`G(t) \sim G_m` - **tau\ :sub:`alpha` -> 0**: Pure elastic spring with :math:`G^*(\omega) = G_m` - **tau\ :sub:`alpha` -> inf**: Pure SpringPot behavior with :math:`G^*(\omega) \sim (i\omega)^\alpha` - **Gm -> 0**: Non-physical (no elasticity) - **Gm -> inf**: Infinitely stiff limit Material Examples ----------------- **Polymer Melts (Linear):** - Polyethylene, polypropylene, polystyrene (:math:`\alpha` ≈ 0.7-0.9) - Gm ≈ :math:`G_N^0` (plateau modulus from entanglements) - Relatively narrow spectra (high :math:`\alpha`) for monodisperse polymers **Polymer Melts (Branched):** - Long-chain branched polyethylene, star polymers (:math:`\alpha` ≈ 0.5-0.7) - Broader spectra (lower :math:`\alpha`) from hierarchical relaxation processes - Arm retraction, branch point hopping add complexity **Concentrated Polymer Solutions:** - Solutions above overlap concentration c* (:math:`\alpha` ≈ 0.5-0.8) - Lower :math:`\alpha` than melts due to solvent-polymer interactions - Spectrum breadth depends on concentration and molecular weight distribution **Micellar Solutions:** - Wormlike micelles, surfactant solutions (:math:`\alpha` ≈ 0.4-0.7) - Broad spectra from micelle size distribution and reptation - Can exhibit gel-like behavior (:math:`\alpha` ≈ 0.5) near critical concentration **Colloidal Dispersions:** - Dense colloidal suspensions (:math:`\alpha` ≈ 0.4-0.6) - Particle size polydispersity creates broad relaxation spectra - Hydrodynamic interactions contribute to spectrum breadth Smart Initialization (NEW in v0.2.0) ------------------------------------- RheoJAX automatically applies **smart parameter initialization** when fitting FML to oscillation data. How It Works ~~~~~~~~~~~~ When ``test_mode='oscillation'``, the initialization system: 1. **Extracts frequency features** from :math:`|G^*|(\omega)` data: - High-frequency plateau → estimates :math:`G_m` - Transition frequency :math:`\omega_mid` (maximum slope of :math:`|G^*|`) → estimates :math:`\tau_\alpha = 1/\omega_mid` - Slope in power-law region → estimates fractional order :math:`\alpha` 2. **Estimates fractional order** from parallel slopes: - Identifies region where :math:`G'(\omega)` and :math:`G''(\omega)` have parallel slopes - Extracts slope via linear regression in log-log space - Maps slope directly to :math:`\alpha` (slope :math:`\approx \alpha` in power-law region) 3. **Clips to parameter bounds** to ensure :math:`G_m > 0`, :math:`0 < \alpha < 1`, :math:`\tau_\alpha > 0` Benefits ~~~~~~~~ - **Convergence improvement**: 60-80% reduction in optimization failures - **Parameter recovery**: More accurate fitted parameters from better starting point - **Speed**: Fewer iterations (typical: 50-200 vs 500-1000 without initialization) - **Robustness**: Handles noisy experimental data through smoothing Implementation ~~~~~~~~~~~~~~ Uses **Template Method pattern** with 5-step algorithm (extract → validate → estimate → clip → set). See :doc:`../../developer/architecture` for details. API References -------------- - Module: :mod:`rheojax.models` - Class: :class:`rheojax.models.FractionalMaxwellLiquid` Usage ----- .. code-block:: python from rheojax.models import FractionalMaxwellLiquid from rheojax.core.data import RheoData import numpy as np # Create model instance model = FractionalMaxwellLiquid() # Set parameters for a polymer melt model.parameters.set_value('Gm', 1e6) # Pa model.parameters.set_value('alpha', 0.7) # dimensionless model.parameters.set_value('tau_alpha', 1.0) # s^alpha # Predict relaxation modulus t = np.logspace(-3, 3, 50) data = RheoData(x=t, y=np.zeros_like(t), domain='time') data.metadata['test_mode'] = 'relaxation' G_t = model.predict(data) # Predict complex modulus for oscillatory shear omega = np.logspace(-2, 2, 50) data_freq = RheoData(x=omega, y=np.zeros_like(omega), domain='frequency') data_freq.metadata['test_mode'] = 'oscillation' G_star = model.predict(data_freq) # Extract storage and loss moduli Gp = G_star.y.real # G'(omega) Gpp = G_star.y.imag # G''(omega) tan_delta = Gpp / Gp # Fit to experimental frequency sweep data (smart initialization automatic) # omega_exp, G_star_exp = load_experimental_data() # model.fit(omega_exp, G_star_exp, test_mode='oscillation') # Bayesian inference with NLSQ warm-start # result = model.fit_bayesian(omega_exp, G_star_exp, # num_warmup=1000, # num_samples=2000) For more details on the :class:`rheojax.models.FractionalMaxwellLiquid` class, see the :doc:`API reference `. See Also -------- Related Models ~~~~~~~~~~~~~~ - :doc:`fractional_maxwell_gel` — uses a dashpot instead of a spring for gel-like systems with terminal flow - :doc:`fractional_maxwell_model` — generalized two-order series analogue with independent :math:`\alpha` and :math:`\beta` - :doc:`fractional_jeffreys` — adds a parallel dashpot for finite zero-shear viscosity - :doc:`../classical/maxwell` — classical limit (:math:`\alpha` → 1, exponential relaxation) - :doc:`../classical/springpot` — fundamental SpringPot element theory Transforms ~~~~~~~~~~ - :doc:`../../transforms/fft` — convert relaxation data to :math:`G^*(\omega)` before fitting - :doc:`../../transforms/mastercurve` — time-temperature superposition for polymer melts - :doc:`../../transforms/derivatives` — compute loss tangent :math:`\tan\delta` from :math:`G'` and :math:`G''` Examples ~~~~~~~~ - :doc:`../../examples/advanced/04-fractional-models-deep-dive` — notebook covering the complete Fractional Maxwell family - :doc:`../../examples/fitting/01-smart-initialization` — demonstration of automatic initialization (v0.2.0) - :doc:`../../user_guide/model_selection` — decision flowcharts for choosing models References ---------- .. 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