Inverse Control of Physically Consistent Terrains – INVTERRA

The methodology integrates computational geomorphology, physics-based simulation, and machine learning into a unified terrain synthesis pipeline. The foundation is a generic terrain evolution model, which encodes geological laws into a structured solver capable of handling multiple erosion and deposition processes. A key feature of this model is the natural ordering of transport processes, which ensures numerical stability and efficiency by structuring computations along gravity-driven flow paths. A significant part of the research is to integrate existing or new erosion laws in this paradigm.

To enable by-example terrain synthesis, we develop a gradient-based inversion framework that estimates spatially varying geological parameters from target topographies. This involves differentiable terrain solvers coupled with geology-informed regularization to prevent overfitting or degenerate solutions. Additionally, the project explores implicit neural representations to learn a generic erosion function directly from real-world datasets. Unlike previous machine-learning approaches, our physics-informed loss functions ensure that learned models respect fundamental conservation laws.

On the computational side, efficiency is achieved through GPU-accelerated flow and depression routing, enabling real-time parameter estimation for interactive terrain control. The optimization framework further incorporates multi-resolution strategies, accelerating convergence by progressively refining the terrain solution from coarse to fine scales. Lastly, a novel preconditioning strategy is developed to improve the conditioning of gradient-based optimization, ensuring stable and efficient inversion of terrain parameters.

INVTERRA introduced novel algorithms for terrain representation and simulation, enhancing computational efficiency and physical realism. We explored the representation of physical law along gravity-driven flow paths applied to a novel erosion model that captures the erosive impact of debris flow – a mixture of mud, water, and rocks – on steep slopes. The efficient solution of these new physical equations requires a specific ordering along the flow paths, for which we proposed first an approximate GPU algorithm before developing an exact version in a library called Fastflow. Meanwhile, we developed a glacial erosion model integrating a deep-learning-based ice-flow emulator with a multi-scale advection scheme, accurately reproducing U-shaped valleys, fjords, and glacial deposits over long geological timescales.

We found that these two contributions - surrogate models for ice dynamics and fast algorithms for flow routing– also contribute to the project’s goals of inverting the physical laws behind landscape evolution. We studied the abilities of our deep-learning emulator for ice dynamics to solve problems in glaciology, such as estimating the bedrock topography from elevation and velocity observations at the glacier surface. Our algorithms for flow routing enabled the development of an “unerosion” model capable of simulating the landscape evolution back in time by reversing the physical equations and the computation of analytical solutions of a fluvial erosion law. This mathematical model replaces costly iterative simulations with a direct, time-controlled terrain aging process, bridging the gap between procedural and physically-based terrain synthesis.

Other outcomes of INVTERRA go beyond terrain generation and extend the proposed methodology to other natural phenomena. In particular, we proposed a simplified model for wind-blown sand transport to predict dune formation and deposition patterns around obstacles, with applications in both environmental and urban planning. Another application targets the simulation of volcanoes, where the combination of simplified physics for the volcanoes plumes and the surrounding atmospheric layers enabled the efficient simulation of important volcanic effects such as ash dispersion, specific cloud formation, and volcanic lightning. Finally, we used Green’s functions as a smoothing kernel for a new lava flow model to enable real-time, large-scale lava simulations while maintaining physical accuracy.

In a world where digital exchanges drive a pressing need for virtual environments, a challenge lies in the authoring of the root of these synthetic worlds: the mountains, plains, and other landforms concatenated and represented as terrains. This problem is notoriously difficult because terrains result from the interplay of physical events over geological time scales. This project aims to explore the inversion of simulation parameters as a novel paradigm for terrain generation in virtual worlds, combining geological consistency, natural diversity, and expressive user control for the first time.