Sampling the parameter subspace for ensemble simulations involves a series of decisions. The number of parameters to vary, their ranges, and their distributions need to be specified. These choices are context-dependent, determined by the prior knowledge of the model setup, the workflow and the study objective. In iterative workflows involving a sensitivity analysis, the parameter set may shrink across iterations, as low influence parameters are identified and removed, requiring the parameter subspace to be redefined at each stage. Additionally, parameters may follow skewed or non-linear distributions (e.g., permeabilities are often defined in logarithmic scale while density or thermal conductivity use a linear scale), requiring sampling strategies that avoid bias in subsequent statistical analysis e.g.,.

The configuration file contains the list of varying parameters, along with the respective value ranges, the ensemble size for training and validation realisation sets, and the sampling strategy for each set. Platypus supports the definition of arbitrary forward and inverse mappings to linearize each parameter distribution prior to sampling, and provides currently two sampling strategies: the Latin Hypercube sampling LHS;, and random sampling. The former is a sampling strategy widely adopted for uniform sampling of the parameter space to construct the training ensemble of models. The latter is often used for the validation ensemble.

Input files are generated from a reference input file template with placeholders for parameter values, ensuring that the ensemble members are built from a verified reference model. The resulting input files can be executed locally or on high-performance compute clusters.

The non-intrusive reduced-basis method is agnostic of the numerical method, making Platypus extensible to outputs from different process-based simulators, though it is currently tailored to GOLEM/MOOSE. GOLEM is an open-source finite element simulator built on the general Multiphysics Object-Oriented Simulation Environment MOOSE;. It implements the coupled thermo-hydro-mechanical formulation describing fluid flow, heat transport, and mechanical deformation in porous media. The coupled system of equations is summarized below.

The governing equation for pore fluid pressure comes from the fluid mass balance, under Boussinesq approximation, as 11MB∂pf∂t-βb∂T∂t+αϵ¯¯˙kk+(1-α)ϵ¯¯˙kk⋆+∇⋅qD=0, where p is the pore pressure, T is the temperature, t is the time, ϵ¯¯˙kk is the total volumetric strain rate, ϵ¯¯˙kk⋆ is the inelastic strain rate, α is the Biot coefficient, MB is the Biot modulus, β is the volumetric thermal expansion coefficient, and the subscripts f, s and b, correspond to the fluid, solid and bulk components, respectively. The Darcy velocity qD is given by 2qD=-1μk¯¯∇pf-ρfg, where k¯¯ is the permeability tensor of the porous medium, μ is the fluid viscosity, ρf is the density of the fluid, and g the gravity vector.

The governing equation for heat transport results from energy balance of the porous-fracture system under thermal equilibrium conditions as 3(ρc)b∂T∂t+∇⋅ρfcfqDT-λb∇T=0, where c is the specific heat capacity, (ρc)b=ϕρfcf+(1-ϕ)ρscs is the bulk specific heat of the porous medium, ϕ is the porosity, λb=ϕλf+(1-ϕ)λs is the bulk thermal conductivity.

The linear momentum balance equation of the bulk in terms of the effective Cauchy stress tensor σ¯¯′ is given by 4∇⋅σ¯¯′-αpfI¯¯+ρbg=0. There, I¯¯ is the rank-two identity tensor, and ρb is the bulk density of the fluid–solid mixture (ρb=ϕρf+(1-ϕ)ρs).

Further details about the governing equations, their derivation from mass, energy, and momentum conservation, and the underlying assumptions can be found in . For a general review of coupled thermo-hydro-mechanical (and chemical) modeling of fractured porous media, see , , and references therein.

Platypus interfaces with a simulator by writing sampled parameter values into the solver's input, and reading the resulting ensemble of model solution snapshots. Supporting a different simulator amounts to adapting how sampled parameters are supplied to the solver, and how output fields are read back into the framework. The rest of the downstream modules remains unaffected. Minimal integration requires a function that exposes the model data to the framework. Deeper integration, such as automated execution of the solver or handling solver-specific mesh structures, requires more effort.

Platypus leverages PyVista and its VTK Exodus readers to read, filter, and visualize GOLEM model outputs for model curation and quantity of interest selection. This allows the user to work with entire model states or restrict the domain to the observation space alone.

The user can filter by predefined mesh block names (e.g., from volumetric lithologies or lower dimensional structures such as faults), cell types (e.g., triangles, tetrahedrons, hexahedrons), fields stored at nodal or cell level (e.g., temperature, pore pressure, stress components), merging the selection into a single unstructured mesh. The resulting reference mesh serves both as visualization output and as common spatial structure across the ensemble.

Model curation at this stage also includes identifying and discarding failed, incomplete or inconsistent simulations prior to ensemble assembly. Once the relevant fields are selected, mesh geometry and connectivity are dropped before packing, retaining only the field data and associated metadata and avoiding data duplication. Spatial information is not required for surrogate training or evaluation under the NIRB approach we employ, though a reference mesh is retained to visualize the spatial distribution of results.

In this stage of data curation, we assemble and store the ensemble of simulations in a multidimensional, labelled data structure. Figure a illustrates the multidimensional character of the stored data, whose axes correspond to physical fields, time values, realization numbers, and spatial indices indicating whether field data was extracted node- or cell-wise. Beyond storage, this step enforces metadata consistency across simulations and provides structured, labelled access to the ensemble data.

Platypus leverages Zarr format with Xarray to store the n-dimensional array, and labels of their values across their axes to maintain interpretability. An additional advantage of using these libraries in the context of big scientific data lies on the distributed, chunked storage, lazy loading, and available compression algorithms for the packed data e.g.,. The distributed storage is well-suited to parallel processing and cloud-native workflows. The lazy loading refers to having access to the ensemble dimensions without loading it all on memory, which is particularly helpful for extracting subdomains of the ensemble prior to loading in memory the data. To accommodate different computational environments, Platypus contains different packing strategies suited to both laptop and high-performance cluster settings.

Platypus applies a static correction and a normalization to improve conditioning and ensure that each feature has equal level of importance to model performance. The static correction, or “particular solution” , is substracted from the data to isolate the perturbations to the system. In Platypus, the user can choose this term as either the inital conditions of the simulation, or the node/cell-wise mean across realizations and time. Platypus supports standarization (Z-score normalization) and min-max-scaling. The computed scaling parameters are stored in JSON format to enable consistent forward and inverse scaling when evaluating the surrogate. The user can select a subset of the ensemble along any of its axes, on which both correction and scaling parameters are computed and the surrogate is trained.

In Platypus we adopt a hybrid surrogate approach, the non-intrusive reduced-basis method . Figure b shows the NIRB method, which consists of a decoupled offline and online phases. In the offline phase, the method uses snapshot collections from the full order models to extract a reduced basis through a proper orthogonal decomposition (POD) to capture characteristic physical behaviour, and uses machine learning to determine the weights of these basis functions. After the offline phase, the online phase is used to obtain efficient on-demand state predictions . This hybrid physics-based machine learning method enables the construction of robust, scalable, and interpretable surrogate models that approximate high-fidelity physical simulations without modifying the governing equations or the underlying solver.

The reduced solution urb for a set of parameters γ is approximated as the linear combination of basis functions, 5urb(γ)=∑i=1nψiΘrb(i)(γ), where ψi are the POD basis functions and Θrb(i) are the corresponding reduced coefficients evaluated for γ. The basis functions are truncated by imposing a threshold ϵ to a cumulative energy criterion defined by the singular values σi as 6∑i=1nσi2∑j=1ntotσj2≥1-ϵ, where ntot is the total number of training snapshots. This criterion selects the n most energetically significant basis functions. In Platypus, this criterion is handled at the configuration file, referred to as POD threshold.

Platypus includes two options to compute the coefficients. The first option is using Gaussian process regression, a non-parametric Bayesian approach that models input-output relations through a normal distribution prior, requiring few hyperparameters and yielding optimal coefficient estimates . It is well-suited to small scenarios and processes with relatively linear parameter variations , with limited scalability (which has motivated efforts to overcome it; ), making it less appropriate for highly non-linear processes. The second option involves estimating the coefficients via neural networks using, in our case, TensorFlow . The latter is rather preferred for coeficient estimation from non-linear processes . This option also features the use of Optuna for hyperparameter tuning . The surrogate training in Platypus adopts the NIRB implementation used and analysed in previous applications . The cost for this offline phase is dependent on the chosen QoI, degrees of freedom and complexity of the field to be reproduced, as well as timesteps and POD threshold, and surrogate training method.

At the online stage, the surrogate model can be evaluated orders of magnitude faster than full-order simulations, enabling both real-time parameter exploration and large-scale statistical analysis. Platypus includes pipelines to compute variance-based Sobol sensitivity indices via SALib . This allows the user to assess the influence of model parameters on selected quantities of interest, helping to reduce the parameter space for applications such as inversion, optimization, and uncertainty quantification. The Sobol method requires no a priori knowledge of parameter distributions and decomposes the model output variance into direct parameter effects (first-order indices), parameter interactions (higher-order indices), and total influence (total-order indices).

While a global sensitivity analysis (SA) is computationally demanding, it is a natural preparation step for model calibration . Taking into consideration the information from parameter interactions significantly reduces the dimensionality of the problem, directly benefiting subsequent calibration and inversion schemes.

The simple reservoir scenario consists of a synthetic geothermal doublet embedded in a low permeability formation at approximately 4 km depth with induced fractures at the two operational wells, designed to capture the thermohydraulic response to geothermal operations. We use this scenario to demonstrate a typical Platypus workflow, covering parameter sampling, model curation, ensemble assembly, surrogate construction, and global sensitivity analysis.

In this transient TH model, we vary porosity, permeability, density, heat capacity, and thermal conductivity of the solid media over ranges specified in Table , while all remaining material parameters are held constant at their reference values. The model domain consists of a 500 m cubic porous reservoir with a background temperature of 150 °C. Water is injected at a temperature of 70 °C through an injection well at a rate of 3 L s⁻¹, with an equal production rate. The scenario has an operation time of 100 years. We sample the training ensemble of 200 realizations via Latin Hypercube sampling, and draw a validation ensemble of 40 realizations by random sampling, applying a logarithmic mapping to the permeability to avoid sampling bias. We run each simulation for 100 years with 100 saved timesteps.

We curate the ensemble by retaining pore pressure and temperature fields node-wise over the full volumetric domain, filtering for tetrahedral cell types and discarding fault surfaces and line elements. We then remove the training set mean from both training and validation sets, and apply standard scaling to the parameter sets and min-max scaling to the field data. POD basis extraction at a tolerance of 10-4 yields 14 basis functions for the pore pressure and 51 for the temperature. We estimate the reduced coefficients via neural networks with hyperparameter optimization through Optuna.

In the online stage, we compare the surrogate predictions against the validation set and use them to accelerate the evaluations for global SA. Figure a–b show a the temperature field after 100 years comparing the full order simulation output and the surrogate prediction. Figure c shows their difference, where blue marks regions where the surrogate under-predicts and red marks regions of over-prediction, with residuals below expected temperature measurement errors. Over all validation realizations, timesteps and spatial degrees of freedom, the relative L2 norm is 9.45×10-4. We then evaluate the surrogate model 1 572 864 times to compute Sobol sensitivity indices, following a Saltelli's sampling scheme using 217 samples, as described in, on the domain averaged temperature, identifying ρdom, and λdom as the most influential parameters by their first-order indices.

The Geothermal Research Platform Groß Schönebeck in Northern Germany is a laboratory for the development of geothermal technologies in hydrothermal reservoir rock with low permeability. Two research wells form a geothermal doublet providing access to water-bearing horizons at depths between 3.9 and 4.4 km at temperatures of 150 °C .

To study the multi-injection stimulation performed at this site, in which large volume of injected water at high flow states triggered micro-seismicity (; chronology and references in ), presented and validated a coupled thermo-hydro-mechanical model of the geothermal site, finding seismic rates sensitivity to fluid-pressure rates, temperature variations, and tectonic conditions.

In this example, we adopt the model from to demonstrate how Platypus can be used to extract specific subdomains and lithological units together with fault features from the full-order model (location shown in Fig. a, b). We further use the extracted subdomain to train a surrogate of the pore pressure field, demonstrating that the workflow generalizes to focused domains and thermo-hydro-mechanical (THM) processes. In contrast to the simple reservoir benchmark, this scenario evolves over timescales of days, emphasizing different dominant physical processes.

The model geometry consists of a 1.6 km × 1.6 km × 0.51 km reservoir. We vary the permeability and porosity for two lithological units listed in Table , maintaining the reference anisotropy ratios for the transversely isotropic permeability, with all remaining parameters held constant at their reference values. This choice of parameter space follows the most influential parameters identified in . We sample a training ensemble of 200 realizations via LHS and a validation set of 50 realizations by random sampling. POD extraction at a tolerance of 5×10-4 results in 28 basis functions for pore pressure and 5 for temperature, and we estimate the reduced coefficients via neural networks with hyperparameter tuning. The relative L2 norm across timesteps, realizations and spatially distributed pore pressure field yields a value of 1.34×10-2. Figure c shows the pore pressure validation, demonstrating that Platypus can be used to construct an accurate surrogate on a spatial subdomain.

The workflow steps in Platypus described above represent a functional but extensible baseline. Several directions for future development are outlined here.

At the sampling stage, the current implementation supports Latin Hypercube and random sampling. Potential extensions include low-discrepancy sequences , as well as adaptive sampling schemes e.g.,, which iteratively refine the parameter space coverage based on model responses.

Adapting Platypus to alternative solvers, such as finite volumes or finite differences codes, would require only a modified model pruning module to extract transient output fields, ensure consistent indexing across the realizations, and assemble the ensemble of simulation data. For simulations that involve adaptive mesh refinement, where the number of nodes varies across time and realizations, projection methods that accommodate mesh variability exist . Alternatively, one may consider projecting onto a common reference mesh solution.

The surrogate training stage is currently based on the NIRB approach with POD for basis extraction, and Gaussian process regression or neural networks for coefficient estimation. This could be extended to alternative reduction methodologies, different machine learning backends, or problem-specific constraints on data handling.

Finally, while Platypus supports an arbitrary number of parameters and output fields, a large parameter space will demand a large ensemble to adequately capture the physical behaviour of the model, which in turn increases the computational cost of both simulation and surrogate training.