We introduce two new concepts of convergence of gradient systems \(({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )\) to a limiting gradient system \(({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)\). These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of \(({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )\). It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences \(({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )\). In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

我们引入梯度系统收敛的两个新的概念\（（{\ mathbf {Q}}，{{\ mathcal {E}}} _ \ varepsilon，{{\ mathcal {R}}} _ \ varepsilon）\）到极限梯度系统\（（{{mathbf {Q}}，{{\ mathcal {E}}} _ 0，{{\ mathcal {R}}} _ 0）\）。这些新概念称为“带倾斜的EDP收敛”和“带倾斜的接触式EDP收敛”。两者均基于梯度系统解决方案的能量耗散原理（EDP）公式，并且可以看作是Sandier和Serfaty首次引入的梯度流Gamma收敛的改进。构造这两个新概念是为了避免有时会作为EDP收敛限制出现的“非自然”限制梯度结构。通过对\（（{\ mathbf {Q}}，{{\ mathcal {E}}} _ \ varepsilon，{{ \ mathcal {R}}} _ \ varepsilon）\）。它避免了不自然的极限梯度结构，但是根据这个概念，许多有趣的系统是不收敛的。带有倾斜的Contact–EDP收敛是带有倾斜的EDP收敛的松弛，并且仍然避免不自然的限制，但适用于更广泛的序列类别\（（{\ mathbf {Q}}，{{\ mathcal {E}}} _ \ varepsilon ，{{\ mathcal {R}}} _ \ varepsilon）\）。在本文中，我们定义了这些概念，研究了它们的属性，并将其与经典EDP收敛联系起来。我们说明了许多测试问题上的不同概念。