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Recent sharpness-aware minimisation (SAM) is known to find flat minima which is beneficial for better generalisation with improved robustness. SAM essentially modifies the loss function by reporting the maximum loss value within the small neighborhood around the current iterate. However, it uses the Euclidean ball to define the neighborhood, which can be inaccurate since loss functions for neural networks are typically defined over probability distributions (e.g., class predictive probabilities), rendering the parameter space non Euclidean. In this paper we consider the information geometry of the model parameter space when defining the neighborhood, namely replacing SAM's Euclidean balls with ellipsoids induced by the Fisher information. Our approach, dubbed Fisher SAM, defines more accurate neighborhood structures that conform to the intrinsic metric of the underlying statistical manifold. For instance, SAM may probe the worst-case loss value at either a too nearby or inappropriately distant point due to the ignorance of the parameter space geometry, which is avoided by our Fisher SAM. Another recent Adaptive SAM approach stretches/shrinks the Euclidean ball in accordance with the scale of the parameter magnitudes. This might be dangerous, potentially destroying the neighborhood structure. We demonstrate improved performance of the proposed Fisher SAM on several benchmark datasets/tasks.