Researchers are increasingly focused on recovering unknown functional components within partial differential equations, a challenge that limits predictive modeling. Torkel E. Loman of the University of Oxford’s Institute of Mathematics, Yurij Salmaniw of Cape Breton University’s Department of Mathematics, Physics and Geology, and Antonio Leon Villares of the University of Oxford’s School of Engineering, in collaboration with colleagues including Jose A. Carrillo and Ruth E. Baker of the University of Oxford’s Institute of Mathematics, demonstrated a new way to learn these functions directly from data. Their work embeds a neural network in a partial differential equation, allowing it to accurately approximate unknown functions during training. By applying this approach to nonlocal agglomeration-diffusion equations, the team successfully recovered interaction kernels and external potentials from steady-state data, while systematically investigating the influence of data quantity, quality, and characteristics on recovery accuracy. This work is important because it extends existing parameter fitting workflows to functional recovery and allows the resulting partial differential equations to be used for prediction of standard systems.
Scientists often encounter partial differential equations that involve unknown functions that are difficult or impossible to measure directly, hindering predictive modeling. If these landscapes can be reconstructed from observations of population density, ecological management decisions can be informed.
In physics and chemical engineering, well-studied inverse problems arise because heat or mass transport systems are ubiquitous and often involve known temperatures or concentrations, but the associated source fields, conductivities, or diffusivities are unknown. A similar role for spatial properties occurs in phase field and pattern formation models, where nonuniform slope or mobility maps encode the effective energy landscape.
In nonlocal agglomerative, diffusive systems, the inferred spatial structure can be an external stimulus or the interaction kernel itself. This discrepancy, of which the identification of classical parabolic sources in heat and transport settings is a well-known example, raises an inverse problem aimed at recovering these unknown spatial components from the data.
Although classical parameter fitting techniques can recover scalar system parameters from data, these can also be extended to recover complete function parameters. This study focused on the agglomeration-diffusion equation and successfully recovered the interaction kernel and external potential directly from steady-state data. The core of this approach lies in minimizing the fixed-point residual, quantified as ∥T(u) − u∥, which vanishes exactly at the numerical equilibrium of the forward model, thereby avoiding differentiation of noisy data.
Initial experiments utilizing accurate and noise-free data confirmed successful recovery of both single and multiple functional parameters. Subsequent analyzes incorporated comprehensive empirical measurements and revealed that recovery is possible even under sparse sampling conditions. However, increasing measurement noise degrades recovery performance, and the extent of this degradation depends on the specific system and data quality.
The ease of functional recovery is highly influenced by factors such as the number of available solutions and their unique characteristics. A more diverse and mutually informative solution facilitates more accurate recovery. Furthermore, recovery failures can be due to lack of structural identifiability, analytically predictable outcomes, or implementation details and data quality. This study establishes the feasibility of functional component recovery, which holds true at a theoretical level using accurate data and remains achievable in real-world scenarios with partial or noisy observations.
big picture
Researchers have demonstrated how to effectively “learn” unknown functions embedded in partial differential equations, going beyond simply identifying numerical parameters to reconstructing the functional relationships governing system behavior. This is not just an improvement on existing technology. It represents a change in the way inverse problems are approached in mathematical modeling.
Scientists have long struggled to derive meaningful insights when key elements of a model cannot be observed. Traditional methods often make simplifying assumptions or rely on extensive and painstaking experimentation. This new work circumvents these limitations by integrating neural networks directly into the partial differential equations themselves, allowing the model to learn missing functions from observed data.
The ability to recover these functions with increased accuracy even with limited or noisy data is particularly noteworthy. The inherent inability to identify the problem remains a major obstacle. This study clearly shows that a single observation is not sufficient to uniquely determine hidden features. To adequately constrain a problem, you need at least two independent data points, or “solutions.”
This highlights the importance of experimental design and data collection strategy. In the future, this technique could be extended to much more complex systems, providing insights in areas ranging from fluid mechanics to materials science and even biological modeling. The next step is not simply to improve the accuracy of recovery, but to develop ways to intelligently sample the solution space and overcome limitations due to lack of data. The ultimate goal is to create models that truly reflect the underlying physical reality, not just predictions.
