MILP initialization for solving parabolic PDEs with PINNs

要約

物理情報に基づいたニューラルネットワーク（PINN）は、物理システムのソリューションとパラメーター推定を提供できる強力な深い学習方法です。
ニューラルネットワーク構造の複雑さを考えると、主に現実的なシステムをモデル化するアプリケーションで使用する場合、収束速度は数値的手法と比較して依然として制限されています。
ネットワークの初期化は、従来のニューラルネットワークの場合のように、初期の重みのランダムな分布に続き、重度のモデル収束ボトルネックにつながる可能性があります。
To overcome this problem, we follow current studies that deal with optimal initial weights in traditional neural networks.
In this paper, we use a convex optimization model to improve the initialization of the weights in PINNs and accelerate convergence.
2つの最適化モデルを最初のトレーニングステップとして調査します。これは、トレーニング前と定義されています。1つは境界と物理学を含む境界のみを含むものです。
The optimization is focused on the first layer of the neural network part of the PINN model, while the other weights are randomly initialized.
We test the methods using a practical application of the heat diffusion equation to model the temperature distribution of power transformers.
境界前トレーニングを備えたPINNモデルは、現在の段階で最も速い収束方法です。

要約(オリジナル)

Physics-Informed Neural Networks (PINNs) are a powerful deep learning method capable of providing solutions and parameter estimations of physical systems. Given the complexity of their neural network structure, the convergence speed is still limited compared to numerical methods, mainly when used in applications that model realistic systems. The network initialization follows a random distribution of the initial weights, as in the case of traditional neural networks, which could lead to severe model convergence bottlenecks. To overcome this problem, we follow current studies that deal with optimal initial weights in traditional neural networks. In this paper, we use a convex optimization model to improve the initialization of the weights in PINNs and accelerate convergence. We investigate two optimization models as a first training step, defined as pre-training, one involving only the boundaries and one including physics. The optimization is focused on the first layer of the neural network part of the PINN model, while the other weights are randomly initialized. We test the methods using a practical application of the heat diffusion equation to model the temperature distribution of power transformers. The PINN model with boundary pre-training is the fastest converging method at the current stage.

arxiv情報

提供元, 利用サービス

arxiv.jp, Google