Data-driven Computational Mechanics

• Data-driven computational mechanics is a model-free approach using experimental stress–strain data to directly impose physical laws via optimization.
• It employs advanced algorithms including max-ent, game-theoretic, and hybrid NLP methods to robustly solve boundary-value problems under uncertainty.
• Practical applications span static/dynamic simulations, hyperelastic and elasto-plastic analyses, and multiscale seismic and biological tissue modeling.

Data-driven computational mechanics (DDCM) constitutes a paradigm shift in computational solid and structural mechanics. It forgoes phenomenological constitutive models, instead formulating boundary-value problems as optimizations or projections that directly reference experimental or high-fidelity simulation data in stress–strain (phase) space. Solutions are obtained by finding admissible field states—satisfying compatibility and equilibrium—that are closest, in a precisely defined sense, to the available material data. Recent research has expanded the theoretical, numerical, and algorithmic foundations of DDCM, explored connections to optimization and game theory, integrated DDCM with machine learning, addressed practical bottlenecks, and demonstrated its potential for rigorously uncertainty-quantified, model-free simulation of nonlinear, stochastic, and inelastic materials.

1. Foundational Principles and Mathematical Formulation

DDCM replaces the constitutive equation $\sigma=\mathcal{C}(\varepsilon)$ with direct use of a material data set $$D_{\text{loc}} = \{(\varepsilon_i, \sigma_i)\}_{i=1}^N$$ sampled from experiment or simulation. The key mathematical structure is the constrained minimization in phase space: $$\text{Find } z \in C,\, \hat{z} \in D \text{ to minimize } d(z,\hat{z}) = \|z - \hat{z}\|\,,$$ where $z = (\varepsilon(x), \sigma(x))$ must satisfy compatibility, equilibrium, and boundary conditions—i.e., $$C = \{ (\varepsilon(x),\sigma(x)) : \varepsilon(x) = \nabla_{\text{sym}} u(x),~ \nabla \cdot \sigma(x) = f(x)$$, and prescribed BCs $\}$. Typically, the distance is evaluated in a metric induced by a compliance-like tensor: $$\| (\varepsilon,\sigma) - (\varepsilon',\sigma') \|^2 = (\varepsilon-\varepsilon'):\mathbb{C}:(\varepsilon-\varepsilon') + (\sigma-\sigma'):\mathbb{C}^{-1}:(\sigma-\sigma')\,.$$ Numerically, this optimization is solved by alternating projections: (i) for given data assignments, solve for an admissible field via a constrained solve; (ii) project the field state at each material point onto the nearest material data. This variational structure guarantees the existence of minimizers, and as the data set “fills in” a classical constitutive manifold, the DDCM solution converges to the conventional one (Kirchdoerfer et al., 2015).

2. Key Algorithmic and Theoretical Variants

Multiple algorithmic, theoretical, and implementation strategies have emerged:

• Distance-minimizing and entropy-maximizing schemes: DDCM can operate either via hard nearest-neighbor assignment or by soft-assignment (Gibbs-measure, “max-ent”) regularization to improve robustness to noise and outliers (Eggersmann et al., 2020, Prume et al., 2022). Max-ent variants weigh all data points exponentially by their (squared) distance to the target state.
• Game-theoretic formulations: Recent work frames DDCM as a non-cooperative game between “strain player” (enforcing admissibility) and “stress player” (enforcing proximity to data). The Nash equilibrium reduces to minimization over an effective data-induced constitutive law and preserves compatibility and equilibrium exactly at each solve (Weinberg et al., 2023).
• Hybrid and nonlinear programming (NLP) approaches: For computational tractability, data can be interpolated into smooth implicit manifolds $g(\varepsilon^*, \sigma^*) = 0$ (via kernel methods or neural networks), leading to a single structured NLP enforcing both physics and data proximity. This alleviates combinatorial cost, allows for implicit/anisotropic laws, and supports kinematic constraints (Gebhardt et al., 2019, Gebhardt et al., 2023).
• Inelasticity and history-dependence: Extensions have incorporated tangent-space augmentation (each data point augmented with $\partial \sigma/\partial \varepsilon$), classification into elastic/inelastic branches, and rule-based data subset transitions—effectively introducing a minimal set of internal variables and transition “memory”, enabling simulation of path-dependent elasto-plastic phenomena (Ciftci et al., 2021, Ciftci et al., 2023).

3. Data Structures, Scalability, and High-Performance Aspects

The dominant computational cost in large-scale DDCM arises from high-dimensional nearest-neighbor (NN) searches over $10^5$–$10^{9}$ data points at every material point and iteration. Advanced data structures and algorithms are critical:

Data Structure	Build Time	Query Cost	Typical Use Case
k-d tree	$O(N \log N)$	exact $O(\log N)$	Low-dimensional (up to $d=12$); moderate $N$
Approx. k-d tree	$O(N \log N)$	$O(1)$–$O(\log N)$	Looser bounds at high $N$; rapid early convergence
k-means tree	Higher	$O(\log N)$	High-dimensional data; large $N$
k-NN graph	High	$O(1)$–$O(\log N)$	Exploit solution locality; large $N$

Empirical studies confirm that, with adaptive scheduling for search “exactness,” speedups of $10^6$ or more over brute-force searches are attainable with negligible impact on solution accuracy (Eggersmann et al., 2020). Further, quantum-inspired swap-test circuits have demonstrated $O(\log D)$ complexity for distance computations (where $D$ is data dimensionality), suggesting potential for quantum-accelerated DDCM pipelines (Xu et al., 2023).

4. Comparison with Conventional and ML-based Approaches

The DDCM paradigm distinguishes itself fundamentally from both classical continuum modeling and supervised machine-learning surrogates:

• Model-free/unsupervised: DDCM does not assume or fit an ansatz for the constitutive law; the “law” emerges directly from the data set via online optimization during each solve (Weinberg et al., 2023).
• Physics integration: Compatibility and equilibrium are enforced at every solve, not by statistical penalty but by constraint satisfaction—the admissible set is given by solution of the field equations.
• Comparison to neural networks (NN): Benchmarks show that in high data-density, low-noise, and in-domain scenarios, DDCM outperforms NNs by $1$–$2$ orders of magnitude in accuracy, but with higher per-query computational cost. NNs, once trained, generalize better to unseen data, larger noise, and geometries outside the training manifold, but incur training cost and possible model bias (Zlatić et al., 27 Aug 2024). Isotropy enforcement and local convex smoothing are essential for robust DDCM performance in the presence of sparse or noisy data.
• Hybrid learning frameworks: Physics-informed neural nets and GANs have been integrated with DDCM to combine the physics-constrained optimization with adversarial learning from data, allowing for “model-free” training of generative models that honor compatibility, equilibrium, and experimental data distributions. These architectures provide stability and data-fidelity in complex, nonlinear problems (Ciftci et al., 2023).

5. Advanced Topics: Stochastic, Dynamic, and Multiscale Extensions

• Stochastic and uncertainty-aware DDCM: The DDCM formalism extends naturally to the inference of distributions under data uncertainty, replacing deterministic solutions with empirical likelihood or convex hull-based upper/lower bounds on responses. Algorithms leveraging Kullback-Leibler minimization, population annealing, and sequential LP yield robust uncertainty quantification even with noise, outliers, or sparse data (Prume et al., 2022, Huang et al., 2022).
• Polyconvexity, anisotropy, and physical consistency: Data-driven models for highly nonlinear anisotropic or biological materials require satisfaction of polyconvexity and monotonicity of energy derivatives. Neural ODE-based architectures can be constructed to guarantee these properties by design, enabling seamless use in large-deformation finite element analyses (Tac et al., 2021).
• Dynamics, dissipation, and multiscale modeling: Variational-integrator-inspired methods have been developed for data-driven dynamics, preserving discrete momentum and supporting implicit, non-potential-based laws (Gebhardt et al., 2019). In wave propagation and site response, data collected from microscale (DEM) simulations are directly used at the macroscale, demonstrating DDCM’s capacity for multiscale coupling and the reproduction of essential dynamic amplification phenomena (Garcia-Suarez et al., 2022).
• Refinement and adaptivity: “Data-refinement” strategies enable efficient multi-regime analysis by selectively applying DDCM only in elements that exceed a linear threshold, marrying the speed of classical FEM with model-free accuracy where required (Wattel et al., 2022).

6. Practical Applications and Numerical Performance

DDCM has been validated in a broad spectrum of benchmark problems and practical contexts:

• Static and dynamic trusses, plates with holes, beams (1D–3D) for both linear and nonlinear constitutive behavior (Kirchdoerfer et al., 2015, Gebhardt et al., 2019, Weinberg et al., 2023).
• Large-deformation hyperelasticity and biological tissue mechanics, capturing intricate nonlinear, anisotropic, and polyconvex responses (Tac et al., 2021).
• Elasto-plasticity and inelasticity with path-dependence, using tangent-space and phase-space augmentation, and exploiting symmetry-based data reduction (e.g., Haigh-Westergaard representation) (Ciftci et al., 2021, Ciftci et al., 2023).
• Seismic soil response, employing DEM-generated mesoscale data for macroscale simulation in 1D site response settings (Garcia-Suarez et al., 2022).
• Fair comparisons with neural networks demonstrate the “narrow but deep” superiority of DDCM with dense, in-domain data, and “broad but shallow” generalization of NNs in the face of noise or data shifts (Zlatić et al., 27 Aug 2024).

Quantitative studies confirm $O(N^{-1/2})$ or $O(N^{-1/d})$ convergence rates with respect to data density, and highlight the paramount importance of data coverage and representativeness. DDI (data-driven identification) can accelerate convergence and reduce data requirements by importance-sampling phase space directly from experimental full-field measurements (Stainier et al., 2019).

7. Limitations, Challenges, and Future Directions

Significant practical and theoretical challenges remain for DDCM:

• Data coverage and noise: The fidelity of the DDCM solution is fundamentally capped by the sampling density and distribution of the data set $D$. Sparse or out-of-domain queries degrade solution accuracy, motivating research into adaptive data enrichment and active learning.
• Computational cost: High-dimensional nearest-neighbor searches and global solves pose challenges in RAM and FLOP for very large meshes or datasets. Efficient data structures, approximate algorithms, and hardware acceleration (including quantum) are active research directions (Eggersmann et al., 2020, Xu et al., 2023).
• Extension to history-dependent, rate-dependent, and multiphysics phenomena: Recent frameworks augment the data set with minimal history variables and transition rules, opening the path to model-free inelastic and viscoelastic simulation, but general comprehensive frameworks are still under development (Ciftci et al., 2021).
• Hybrid-optimization-based formulations: Interpolative or regression-based surrogate manifolds, constructed via neural networks or kernel methods, trade strict model-free purity for computational efficiency and tractability in high dimensions (Gebhardt et al., 2023, Gebhardt et al., 2019).
• Contact, inequality constraints, and complex boundaries: Recent formulations have recast DDCM as mathematical programs with complementarity constraints (MPCC) to enable efficient simulation with contact and friction, using specialized heuristic solvers (Gebhardt et al., 2023).
• Best practices for real-world applications: Integration with standard FEM codes, strategies for data acquisition, robust error estimation, and certification for engineering deployment are highly active areas.

The DDCM paradigm thus continues to expand the interface between data, computation, and continuum physics, enabling model-free, physically consistent, and uncertainty-quantified computational mechanics across a growing range of applications and material behaviors.