Conservative Data-Driven FEM Framework

• The paper introduces a finite element framework that replaces constitutive laws with empirical data to rigorously enforce conservation laws.
• It leverages both primal and mixed finite element schemes to maintain equilibrium and enable intrinsic uncertainty quantification.
• The approach utilizes adaptive error indicators and nonlinear fixed-point iterations to robustly handle heterogeneous, noisy datasets in simulations.

A conservative data-driven finite element framework is an approach to numerical simulation in which conservation laws and boundary conditions are imposed via the finite element method (FEM), while material behavior is integrated directly from empirical or synthetic data, rather than an explicit constitutive law. This paradigm enables simulations that leverage heterogeneous, possibly noisy, experimental datasets, promoting model-free fidelity and robust prediction even in complex, nonlinear, or poorly characterized materials. Recent advances in this field include the development of both primal (stress-strain) and mixed (flux-based) finite element schemes that ensure strong conservation properties, rigorously enforce equilibrium, and supply a posteriori uncertainty quantification that is intrinsic to the data-driven formulation (Korzeniowski et al., 2021, Kuliková et al., 22 Jun 2025).

1. Mathematical Formulation and Fundamental Concepts

The data-driven finite element methodology operates by formulating the material response as a minimization problem in the space of admissible fields, subject to the constraints imposed by physical laws. The canonical form, exemplified in finite-strain elasticity, seeks pairs of strains $\epsilon$ and stresses $\sigma$ over a discretized domain $\Omega$:

$$\min_{(\epsilon(x),\,\sigma(x))} \;\sum_{e=1}^N |\Omega_e|\,d^2((\epsilon^e,\sigma^e),\,D)$$

subject to

$$\nabla\cdot\sigma + b = 0 \quad\text{in}\;\Omega,\qquad \epsilon = \tfrac12(\nabla u + \nabla u^\top),\quad u|_{\partial\Omega_u} = u_0,\;\; \sigma\cdot n|_{\partial\Omega_t}=t_0$$

Here, $d^2((\epsilon,\sigma), D)$ denotes the squared distance of the strain–stress pair to the material dataset $D$ in an appropriate norm, generally constructed using a block metric involving a reference elasticity tensor $C$:

$$\left\|(\epsilon,\sigma)\right\|_C^2 = \epsilon:C:\epsilon + \sigma:C^{-1}:\sigma,$$

with $C^{-1}$ the compliance tensor. Thus, the method is strictly model-free: the database $D$—often sampled from representative volume elements (RVEs)—supplants any closed-form constitutive law (Korzeniowski et al., 2021).

In the context of mixed formulations, the conservation law (e.g., energy, mass, or momentum) is enforced in either strong or weak form:

• Strong form: $\nabla\cdot\mathbf{q} = s$ in $\Omega$
• Weaker (mixed) form: Find flux $\mathbf q\in H(\mathrm{div};\Omega)$ and scalar Lagrange multiplier $\lambda\in L^2(\Omega)$ so that for all $\delta\lambda \in L^2(\Omega)$,

$$\int_\Omega \delta\lambda\left(\nabla\cdot \mathbf{q} - s\right)\, d\Omega = 0.$$

This formulation enforces strong conservation (continuity of normal flux) across element boundaries (Kuliková et al., 22 Jun 2025).

2. Function Spaces, Approximation, and Regularity

A distinguishing feature of the conservative data-driven mixed FEM is the choice of approximation spaces for primary and flux variables. There are two principal configurations (Kuliková et al., 22 Jun 2025):

Formulation	Field regularity	Admissible spaces
Stronger mixed	$T\in H^1(\Omega)$, $\mathbf q\in \mathbf L^2(\Omega)$	$V_h^T=\{T_h\in H^1: T_h=\bar T\}$, $\Sigma_h = \mathbf{L}^2$
Weaker mixed	$T\in L^2(\Omega)$, $\mathbf q\in H(\mathrm{div})$	$V_h^T=L^2$, $$\Sigma_h =\{\mathbf q_h\in H(\mathrm{div}): \mathbf q_h\cdot \mathbf n = \bar q\}$$

The "weaker" mixed choice relaxes the requirements on the primary variable (e.g., temperature or displacement) to $L^2$—allowing discontinuities across element faces—while enforcing flux continuity via the $H(\mathrm{div})$ conformity of $\mathbf{q}$. This is essential for maintaining the conservation law in the strong sense even when data or solution regularity is limited.

The practical implication is that the framework is well-suited for applications with imperfect, incomplete, or noisy data, and supports computational adaptivity (in both mesh and polynomial degree).

3. Integration of Material Data and Solution Algorithms

Material data is represented as a finite (possibly large) set of tuples sampled from direct experiment, multiscale simulation, or synthetic sources (such as RVE-generated foam stress-strain data (Korzeniowski et al., 2021) or heat-flux-gradient-temperature triplets in nonlinear heat conduction (Kuliková et al., 22 Jun 2025)). At each spatial integration (Gauss) point, material consistency is realized by minimizing the distance to the nearest data point:

$$d\{T, \mathbf{g}, \mathbf{q}; \mathcal{D}\} = \min_{(T^*, \mathbf{g}^*, \mathbf{q}^*) \in \mathcal{D}} \;\sqrt{S_T (T-T^*)^2 + S_g \|\mathbf{g} - \mathbf{g}^*\|^2 + S_q \|\mathbf{q} - \mathbf{q}^*\|^2}$$

where the scales $S_T, S_g, S_q$ normalize dimensional contributions.

Algorithmically, the minimization–projection problem is generally solved via nonlinear fixed-point (alternating projection) iterations:

1. Local data search: For each finite element or integration point, perform a nearest-neighbor search in the dataset.
2. Global projection: Solve the finite element balance equations (compatibility, equilibrium, and conservation constraints) with the locally projected data values.
3. Convergence: The iterative process continues until the global average distance to the data and solution changes fall below prescribed thresholds.

This method is proven to converge to a stationary point of the constrained minimization given mild conditions on the dataset and metric tensor (Korzeniowski et al., 2021, Kuliková et al., 22 Jun 2025).

4. Adaptivity, Uncertainty Quantification, and Error Indicators

The conservative data-driven FEM paradigm supports comprehensive a posteriori assessment of solution quality and uncertainty that is intrinsic to the data-centric formulation. Two principal strategies are employed (Kuliková et al., 22 Jun 2025):

• Adaptive $hp^*$-refinement: Error indicators guide local refinement of mesh size ($h$) and polynomial degree ($p$), allocating computational resources to regions of highest FEM error and data uncertainty. Indicators include:
  • Gradient-mismatch $\eta_e = \|\nabla T - \mathbf g\|_{L^2(\Omega_e)}$
  • Flux-conservation $$\nu_e = h_e \|s - \nabla\cdot\mathbf{q}\|_{L^2(\Omega_e)}$$
  • Temperature-jump $$\gamma_e = \sum_{F\subset\partial\Omega_e} h_F^{-1/2} \|[T]\|_{L^2(F)}$$
  • Data-driven consistency and variability ($d^e_{\text{ave}}$, $d^e_{\text{std}}$)
• Uncertainty Quantification (UQ): The "solution manifold" is probed for non-uniqueness that arises from gaps or noise in the dataset via Markov chain Monte Carlo (MCMC) sampling. Random perturbations are applied to the field variables, and each realization is recomputed, yielding sample means and variances for fields of interest. The spatial standard deviation fields $$\sigma_T(\mathbf{x}),\sigma_{\mathbf{q}}(\mathbf{x})$$ identify locations of solution ambiguity induced by data limitations.

This dual-adaptivity is central to the robust deployment of data-driven FEM in realistic scenarios where "data holes" may exist or experimental errors are significant.

5. Benchmarking and Conservation Properties

Empirical studies validate the conservative properties of data-driven FEM algorithms. For example, in the context of elasticity:

• For 1D bar problems, the conservative DDFEM framework exactly preserves path-independence and admits no artificial dissipation when the dataset $D$ is consistent with a 1D constitutive curve, as evidenced by vanishing energy differences in cyclic loading up to solver tolerances (Korzeniowski et al., 2021).
• In full 3D benchmarks (e.g., open-cell foam seals under large deviatioric strain up to $50\%$), the solution tracks the nonlinear microstructural response encoded by the dataset without spurious energy loss, remaining stable and accurate even when a traditional hyperelastic model would fail to capture local fluctuations or perform robustly with noisy data.

Similarly, in the nonlinear heat transfer demonstration on nuclear graphite (Kuliková et al., 22 Jun 2025), the adaptive $hp^*$-strategy localizes computational effort and uncertainty quantification to regions with missing or severely corrupted synthetic dataset points, reflecting the inherent linkage between dataset integrity and prediction reliability.

6. Material Data Generation and Structural Integration

The success of the conservative data-driven FEM relies on the systematic generation of robust, physically representative datasets. For open-cell foams, this is realized by constructing a set of RVEs using methods such as Laguerre tessellation based on μCT scans for geometric realism. Each RVE is subjected to canonical load paths (tension, shear, volumetric), and the macroscopic averages of strain and stress are computed via standard FEM solves at the microscale. The ensemble thus constructed typically comprises $10^3$–$10^4$ data points, which are then employed as the reference material database $D$ in macroscopic computations (Korzeniowski et al., 2021).

For heat diffusion, a synthetic database may sample numerous temperature gradients and resulting heat fluxes across patches, with deliberate injection of missing or noisy data to benchmark algorithmic robustness (Kuliková et al., 22 Jun 2025). This approach generalizes to other multi-physics situations, provided the dataset exhaustively spans the relevant response space.

7. Scope, Limitations, and Perspectives

The conservative data-driven FEM represents a departure from classical constitutive modeling, directly integrating experimental or synthetic material data into the solution process. Its strengths include: avoidance of model bias, rigorous enforcement of physical constraints, quantifiable uncertainty induced by data limitations, and adaptive algorithmic control.

However, the framework is computationally intensive due to repeated nearest-neighbor searches and fixed-point iterations, particularly as dataset size grows. The predictive accuracy is inherently linked to the coverage and fidelity of the underlying dataset; "data holes" or noise directly propagate to solution non-uniqueness and uncertainty. The methodology is extensible to coupled-physics problems and forms an essential foundation for digital twin technologies in complex engineering systems (Kuliková et al., 22 Jun 2025).

A plausible implication is that conservative data-driven FEM approaches will see increasing adoption in fields requiring high-fidelity, model-free simulation with transparent uncertainty quantification, particularly where material models are poorly characterized or the available data is stochastic and incomplete.