# Better-conditioned Inverse Problems in Computational Materials Science

*The original machine learning model for an interatomic potential shown in blue on the bottom right is underconstrained. When trained on QM reference data (green) with machine learning the model produces a wide range of possible predictions for quantities of interest such as the vacancy formation energy. In this project we will explore approaches to regularise these models with improved priors that take account of physical constraints and experimental observations (orange), leading to predictions in better agreement with the available data (orange dashed line).*

**Supervisors:** Prof. James Kermode (Eng.) and Dr. Thomas Hudson (Maths)

**Partners:** Dr. Tom Swinburne (CiNAM Marseille, CNRS, France) and Prof. Christoph Ortner (UBC, Canada)

**Summary: **

**Inverse problems** are a general class of problems that involve calibrating the parameters of a model using measurements of its outputs, typically from real-world experiments. Many such problems occur across computational science, e.g. in the calibration of constitutive parameters such as elastic moduli on the basis of simulations. This PhD project will tackle inverse problems in compuational materials science the framework of **Machine Learning Interatomic Potentials** (MLIPs). Inverse problems are often mathematically ill-posed, meaning there is no single, stable, well-defined solution. This issue may be resolved numerically either using classical optimisation approaches which select a single solution (that may be an artefact of the choice of optimizer) or using tools from **computational** **statistics **and** machine learning** such as Bayesian inference which mitigate the ill-conditioning of the problem by incorporating prior information.

**Background**

Many **machine-learning models for interatomic interactions** have been proposed recently: together, these allow flexible descriptions of atomic environments [1]. This flexibility comes with the challenge of needing to choose parameters for these models that accurately describe complex material processes and produce predictions which agree with experimental observations. A promising route to tackling inverse problems efficiently is through **end-to-end differentiable simulations** (e.g. jax-md, Molly.jl), where the final output quantity of interest can be differentiated with respect to the model parameters. This enables rapid optimisation of and sampling over model parameters to match available reference data.

In this PhD project you will build on the **atomic cluster expansion** (ACE) approach (e.g. using the ACEpotentials.jl [2] or MACE [3,4] codes) to tackle inverse problems. This approach is attractive for inverse problems as it provides a complete basis set for atomic environments; incorporation of this basis in linear models [5] gives rise to analytically tractable uncertainty estimates on output quantities of interest (cf. Iain Best’s HetSys PhD project).

As a first goal, linear ACE models will be trained to predict simple material properties such as elastic constants, with the goal of producing improved priors that restrict models to realistic ranges of the target property. The initial focus will be on single-component materials where there is no internal relaxation, later moving to multi-component materials and impurities. The project will be extended to more complex quantities of interest (detailed in project outcomes below).

**Expected Outcomes**

The PhD project will produce improved priors for MLIPs that will allow targeted material properties to be controlled for the first time. It will also allow the incorporation of observations (e.g. from experiment) to constrain both parameters and model architectures.

Sources of uncertainty (i.e. unknown inputs to forward models) include

- Variation in DFT training data (e.g. due to unknown XC functional)
- Model form error in interatomic potential
- Parametric error in interatomic potential parameters
- Limited training data and choice of configurations/complexity

Example quantities of interest (experimental observables, inputs to inverse problem) include

- Elastic properties
- Solute properties - e.g. segregation, solubility
- Thermally activated transport properties (e.g. point defect migration rates)
- Finite temperature properties - thermal expansion, thermal conductivity

**Links to HetSys training**

The student will benefit from training in materials modelling across the scales (PX911 and PX912), research software engineering (PX913) and predictive modelling and uncertainty quantification (PX914). The new software to be developed will likely be implemented within the ACEpotentials.jl or MACE codebases which use best-practice scientific software development techniques. Some quantities of interest will make use of the matscipy Python package co-developed by Kermode’s group, to which the student may also contribute.

A robust description of uncertainties in the forward model is critical to solve inverse problems. Here this will be based on conformal prediction, using methodology developed by Iain Best in his PhD, which lifts the limitations of purely Bayesian approaches to quantify parametric uncertainty that don’t account for model-form error. The project is also closely aligned with cutting-edge developments in scientific machine learning (SciML), since inverse problems require a fusion of mechanistic modelling in the forward model with experimental data. The ACE basis is also at the forefront of current SciML methodologies for describing interatomic environments.

**Relevant references:**

##### Are you interesting in applying for this project? Head over to our Study with Us page for information on the application process, funding, and the HetSys training programme

At the University of Warwick, we strongly value equity, diversity and inclusion, and HetSys will provide a healthy working environment, dedicated to outstanding scientific guidance, mentorship and personal development.

HetSys is proud to be a part of the Engineering Department which holds an Athena SWAN Silver award, a national initiative to promote gender equality for all staff and students.