Finally,thepredictedoutput function is decoded by the inverse PhCA, given any query locations.

We consider the task of solving inverse problems.

We propose a Gaussian Process (GP)-based policy iteration framework for addressing both forward and inverse problems in Hamilton--Jacobi--Bellman (HJB) equations and mean field games (MFGs). Policy iteration is formulated as an alternating procedure between solving the value function under a fixed control policy and updating the policy based on the resulting value function. By exploiting the linear structure of GPs for function approximation, each policy evaluation step admits an explicit closed-form solution, eliminating the need for numerical optimization. To improve convergence, we incorporate the additive Schwarz acceleration as a preconditioning step following each policy update. Numerical experiments demonstrate the effectiveness of Schwarz acceleration in improving computational efficiency.

We consider the task of solving inverse problems.

Bayesian inference for the truncated GP can be achieved, in many cases with substantially lower computation than would be required using MCMC methods.

We parametrically solve the Boltzmann equations governing freeze-in dark matter (DM) in alternative cosmologies with Physics-Informed Neural Networks (PINNs), a mesh-free method. Through inverse PINNs, using a single DM experimental point -- observed relic density -- we determine the physical attributes of the theory, namely power-law cosmologies, inspired by braneworld scenarios, and particle interaction cross sections. The expansion of the Universe in such alternative cosmologies has been parameterized through a switch-like function reproducing the Hubble law at later times. Without loss of generality, we model more realistically this transition with a smooth function. We predict a distinct pair-wise relationship between power-law exponent and particle interactions: for a given cosmology with negative (positive) exponent, smaller (larger) cross sections are required to reproduce the data. Lastly, via Bayesian methods, we quantify the epistemic uncertainty of theoretical parameters found in inverse problems.

In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the variety of architectures explored in previous works, several open questions still remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of-the-art (SOTA) performance for forward and inverse problems on a wide range of synthetic datasets, and also investigate their zero-shot effectiveness on two real-world-like datasets.

The inverse medium problem, inherently ill-posed and nonlinear, presents significant computational challenges. This study introduces a novel approach by integrating a Neumann series structure within a neural network framework to effectively handle multiparameter inputs. Experiments demonstrate that our methodology not only accelerates computations but also significantly enhances generalization performance, even with varying scattering properties and noisy data. The robustness and adaptability of our framework provide crucial insights and methodologies, extending its applicability to a broad spectrum of scattering problems. These advancements mark a significant step forward in the field, offering a scalable solution to traditionally complex inverse problems.

We propose a unified framework for delay differential equations (DDEs) based on deep neural networks (DNNs) - the neural delay differential equations (NDDEs), aimed at solving the forward and inverse problems of delay differential equations. This framework could embed delay differential equations into neural networks to accommodate the diverse requirements of DDEs in terms of initial conditions, control equations, and known data. NDDEs adjust the network parameters through automatic differentiation and optimization algorithms to minimize the loss function, thereby obtaining numerical solutions to the delay differential equations without the grid dependence and polynomial interpolation typical of traditional numerical methods. In addressing inverse problems, the NDDE framework can utilize observational data to perform precise estimation of single or multiple delay parameters, which is very important in practical mathematical modeling. The results of multiple numerical experiments have shown that NDDEs demonstrate high precision in both forward and inverse problems, proving their effectiveness and promising potential in dealing with delayed differential equation issues.